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+  
+
+  
+
+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
+  
 
+  <p class="powered-by footer-license-icons">
+    <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank" aria-label="Creative Commons">
+      <i class="fab fa-creative-commons fa-2x" aria-hidden="true"></i>
+      <i class="fab fa-creative-commons-by fa-2x" aria-hidden="true"></i>
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/mermaid/8.4.8/mermaid.min.js" integrity="sha256-lyWCDMnMeZiXRi7Zl54sZGKYmgQs4izcT7+tKc+KUBk=" crossorigin="anonymous" title="mermaid"></script>
+        <i class="fab fa-creative-commons-nc fa-2x" aria-hidden="true"></i>
       
-
       
-        
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/highlight.min.js" integrity="sha256-eOgo0OtLL4cdq7RdwRUiGKLX9XsIJ7nGhWEKbohmVAQ=" crossorigin="anonymous"></script>
-        
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/languages/julia.min.js"></script>
-        
+        <i class="fab fa-creative-commons-nd fa-2x" aria-hidden="true"></i>
       
+    </a>
+  </p>
 
-    
 
-    
-    
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.js" integrity="sha256-EErZamuLefUnbMBQbsEqu1USa+btR2oIlCpBJbyD4/g=" crossorigin="anonymous"></script>
-    
 
-    
-    
-    <script>const code_highlighting = true;</script>
-    
 
-    
-    
-    <script>const isSiteThemeDark = false;</script>
-    
 
+  <p class="powered-by">
     
     
     
+      
+      
+      
+      
+      
+      
+      Published with <a href="https://hugoblox.com/?utm_campaign=poweredby" target="_blank" rel="noopener">Hugo Blox Builder</a> — the free, <a href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">open source</a> website builder that empowers creators.
     
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-    <script>
-      const search_config = {"indexURI":"/index.json","minLength":1,"threshold":0.3};
-      const i18n = {"no_results":"No results found","placeholder":"Search...","results":"results found"};
-      const content_type = {
-        'post': "Posts",
-        'project': "Projects",
-        'publication' : "Publications",
-        'talk' : "Talks"
-        };
-    </script>
-    
+  </p>
+</footer>
 
+    </div>
     
-    
+  </div>
 
-    
-    
-    <script id="search-hit-fuse-template" type="text/x-template">
-      <div class="search-hit" id="summary-{{key}}">
+  
+
+
+<script src="/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js"></script>
+
+
+
+
+  
+
+  
+  
+
+  
+
+
+
+
+
+
+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
+
+
+
+
+
+
+
+
+  
+  <script id="search-hit-fuse-template" type="text/x-template">
+    <div class="search-hit" id="summary-{{key}}">
       <div class="search-hit-content">
         <div class="search-hit-name">
           <a href="{{relpermalink}}">{{title}}</a>
@@ -634,42 +899,18 @@ <h2>Latest</h2>
           <p class="search-hit-description">{{snippet}}</p>
         </div>
       </div>
-      </div>
-    </script>
-    
+    </div>
+  </script>
+  
+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
+    <script src="https://cdn.jsdelivr.net/gh/julmot/mark.js@8.11.1/dist/jquery.mark.min.js" integrity="sha512-mhbv5DqBMgrWL+32MmsDOt/OAvqr/cHimk6B8y/bx/xS88MVkYGPiVv2ixKVrkywF2qHplNRUvFsAHUdxZ3Krg==" crossorigin="anonymous"></script>
+  
 
-    
-    
-    <script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/3.2.1/fuse.min.js" integrity="sha256-VzgmKYmhsGNNN4Ph1kMW+BjoYJM2jV5i4IlFoeZA9XI=" crossorigin="anonymous"></script>
-    <script src="https://cdnjs.cloudflare.com/ajax/libs/mark.js/8.11.1/jquery.mark.min.js" integrity="sha256-4HLtjeVgH0eIB3aZ9mLYF6E8oU5chNdjU6p6rrXpl9U=" crossorigin="anonymous"></script>
-    
 
-    
-    
 
-    
-    
 
-    
-    
 
-    
-    
-    
-    
-    
-    
-    
-    
-    
-      
-    
-    
-    
-    
-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
-    
 
 
 
@@ -678,31 +919,53 @@ <h2>Latest</h2>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
-    <a href="https://gohugo.io" target="_blank" rel="noopener">Hugo</a>.
 
-    
-    <span class="float-right" aria-hidden="true">
-      <a href="#" class="back-to-top">
-        <span class="button_icon">
-          <i class="fas fa-chevron-up fa-2x"></i>
-        </span>
-      </a>
-    </span>
-    
-  </p>
-</footer>
 
-  </div>
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
   
+  
+
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
 
   
 <div id="modal" class="modal fade" role="dialog">
@@ -715,7 +978,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -730,5 +994,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 </body>
 </html>
diff --git a/_headers b/_headers
new file mode 100644
index 0000000..48ad1b4
--- /dev/null
+++ b/_headers
@@ -0,0 +1,15 @@
+# Netlify headers
+# Automatically generated
+# Documentation: https://hugoblox.com/docs/hugo-tutorials/security/
+/*
+  
+  X-XSS-Protection: 1; mode=block
+  X-Content-Type-Options: nosniff
+  Strict-Transport-Security: max-age=31536000; includeSubDomains
+  Referrer-Policy: strict-origin-when-cross-origin
+  
+  Permissions-Policy: accelerometer=(), camera=(), geolocation=(), gyroscope=(), magnetometer=(), microphone=(), payment=(), usb=()
+/index.webmanifest
+  Content-Type: application/manifest+json
+/index.xml
+  Content-Type: application/rss+xml
diff --git a/_redirects b/_redirects
new file mode 100644
index 0000000..0797697
--- /dev/null
+++ b/_redirects
@@ -0,0 +1,3 @@
+# Netlify redirects
+# Automatically generated
+# Documentation: https://docs.netlify.com/routing/redirects/
\ No newline at end of file
diff --git a/authors/admin/avatar.png b/authors/admin/avatar.png
new file mode 100644
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diff --git a/authors/admin/avatar_hu5bdd272e648badb42a595cb37d3ba435_2457515_270x270_fill_lanczos_center_3.png b/authors/admin/avatar_hu5bdd272e648badb42a595cb37d3ba435_2457515_270x270_fill_lanczos_center_3.png
new file mode 100644
index 0000000..e2cf1a3
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diff --git a/authors/admin/avatar_old.png b/authors/admin/avatar_old.png
new file mode 100644
index 0000000..5c3cad7
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diff --git a/categories/index.html b/categories/index.html
index 8321ff0..dd3fbfd 100644
--- a/categories/index.html
+++ b/categories/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/categories/">
+
+
+
+
+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
-  <script src="/js/mathjax-config.js"></script>
   
 
   
+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
+
   
   
   
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/academicons/1.8.6/css/academicons.min.css" integrity="sha256-uFVgMKfistnJAfoCUQigIl+JfUaP47GrRKjf6CTPVmw=" crossorigin="anonymous">
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-
     
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1.9.4/css/academicons.min.css" integrity="sha512-IW0nhlW5MgNydsXJO40En2EoCkTTjZhI3yuODrZIc8cQ4h1XcF53PsqDHa09NqnkXuIe0Oiyyj171BqZFwISBw==" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
+
     
     
-      
     
     
       
       
-        
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/github.min.css" crossorigin="anonymous" title="hl-light">
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/dracula.min.css" crossorigin="anonymous" title="hl-dark" disabled>
-        
-      
     
-
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.css" integrity="sha256-SHMGCYmST46SoyGgo4YR/9AlK1vf3ff84Aq9yK4hdqM=" crossorigin="anonymous">
     
 
     
-
     
     
-      
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +141,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,191 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
+  
+  
+  
+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
+
+  
+  
   
 
   
   
   
   
-  <link rel="stylesheet" href="/css/academic.css">
+  
+  
+  
+    
+    
+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
+  
+
+  
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
 
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
   
+  
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
+
 
 
 
 
   
 
+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
+
+
+
+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/categories/" />
+<link rel="canonical" href="https://frankschae.github.io/categories/" />
+
+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
+
+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
+<link rel="apple-touch-icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_180x180_fill_lanczos_center_3.png" />
+
+<meta name="theme-color" content="#1565c0" />
+
+
+
+
+
+
+
+
+
 
   
   
 
+
+
+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
+<meta property="twitter:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" />
+
+
+
   
-  <link rel="alternate" href="/categories/index.xml" type="application/rss+xml" title="FS">
+
+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/categories/" />
+<meta property="og:title" content="Categories | FS" />
+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
+
   
 
-  <link rel="manifest" href="/index.webmanifest">
-  <link rel="icon" type="image/png" href="/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_2.png">
-  <link rel="apple-touch-icon" type="image/png" href="/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_192x192_fill_lanczos_center_2.png">
 
-  <link rel="canonical" href="https://frankschae.github.io/categories/">
+
+
+
+
+
+
+
 
   
   
   
+
   
   
+    <link rel="alternate" href="/categories/index.xml" type="application/rss+xml" title="FS" />
   
+
   
-    
-    
+
+
   
+  <title>Categories | FS</title>
+
   
-  <meta property="twitter:card" content="summary">
   
-  <meta property="og:site_name" content="FS">
-  <meta property="og:url" content="https://frankschae.github.io/categories/">
-  <meta property="og:title" content="Categories | FS">
-  <meta property="og:description" content="Postdoctoral researcher"><meta property="og:image" content="https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png">
-  <meta property="twitter:image" content="https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png"><meta property="og:locale" content="en-us">
   
-    
   
 
-  
 
 
 
 
-  
 
 
-  
 
 
 
 
+</head>
+
 
-  <title>Categories | FS</title>
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
 
-</head>
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
+
+  
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<aside class="search-modal" id="search">
   <div class="container">
     <section class="search-header">
 
@@ -275,17 +399,23 @@
           <h1>Search</h1>
         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +428,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
+  
   
 
 
@@ -306,153 +442,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    <button type="button" class="navbar-toggler" data-toggle="collapse"
-            data-target="#navbar-content" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
-    <span><i class="fas fa-bars"></i></span>
-    </button>
-    
 
-    
-    <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    
-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
-        
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
 
-        
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
-        
-        
+        <ul class="navbar-nav d-md-inline-flex">
           
-        
 
-        
-        
-        
-        
-        
-        
           
+
           
           
-            
           
+            
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
-        
-        
-        
           
-        
+          
 
-        
-        
-        
-        
-        
-        
           
+
           
           
           
-        
+            
+          
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#software"><span>Open Source Software</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
+          
+          
+
+          
+
+          
+          
+          
+            
+          
+
+          
+
+          
+          
+          
+          
+
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
 
         
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
@@ -461,65 +679,51 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
+
         
         
-          
-          
-          
-            
-          
-          
         
-
-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
+            </a>
+          </div>
         </li>
+        
 
         
         
 
-      
-
-        
       </ul>
-    </div>
-
-    <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
-      
-      <li class="nav-item">
-        <a class="nav-link js-search" href="#"><i class="fas fa-search" aria-hidden="true"></i></a>
-      </li>
-      
-
-      
-      <li class="nav-item dropdown theme-dropdown">
-        <a href="#" class="nav-link js-theme-selector" data-toggle="dropdown" aria-haspopup="true">
-          <i class="fas fa-palette" aria-hidden="true"></i>
-        </a>
-        <div class="dropdown-menu">
-          <a href="#" class="dropdown-item js-set-theme-light">
-            <span>Light</span>
-          </a>
-          <a href="#" class="dropdown-item js-set-theme-dark">
-            <span>Dark</span>
-          </a>
-          <a href="#" class="dropdown-item js-set-theme-auto">
-            <span>Automatic</span>
-          </a>
-        </div>
-      </li>
-      
 
-      
+    </div>
+  </nav>
+</header>
 
-    </ul>
 
   </div>
-</nav>
 
+  <div class="page-body">
+    
+    
+    
+
+    
 
-  
 
 
 
@@ -552,77 +756,144 @@ <h1>Categories</h1>
 <div class="universal-wrapper">
   
 
+
+  
+
   
   
 
   
 
-</div>
+  
 
-      
+</div>
+  </div>
 
+  <div class="page-footer">
+    
     
+    <div class="container">
+      <footer class="site-footer">
+
+  
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+  
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+
+  
+  
     
+  
+  
     
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script>
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/jquery.imagesloaded/4.1.4/imagesloaded.pkgd.min.js" integrity="sha256-lqvxZrPLtfffUl2G/e7szqSvPBILGbwmsGE1MKlOi0Q=" crossorigin="anonymous"></script>
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/jquery.isotope/3.0.6/isotope.pkgd.min.js" integrity="sha256-CBrpuqrMhXwcLLUd5tvQ4euBHCdh7wGlDfNz8vbu/iI=" crossorigin="anonymous"></script>
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+  
+
+  
+
+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
+  
 
+  <p class="powered-by footer-license-icons">
+    <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank" aria-label="Creative Commons">
+      <i class="fab fa-creative-commons fa-2x" aria-hidden="true"></i>
+      <i class="fab fa-creative-commons-by fa-2x" aria-hidden="true"></i>
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/mermaid/8.4.8/mermaid.min.js" integrity="sha256-lyWCDMnMeZiXRi7Zl54sZGKYmgQs4izcT7+tKc+KUBk=" crossorigin="anonymous" title="mermaid"></script>
+        <i class="fab fa-creative-commons-nc fa-2x" aria-hidden="true"></i>
       
-
       
-        
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/highlight.min.js" integrity="sha256-eOgo0OtLL4cdq7RdwRUiGKLX9XsIJ7nGhWEKbohmVAQ=" crossorigin="anonymous"></script>
-        
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/languages/julia.min.js"></script>
-        
+        <i class="fab fa-creative-commons-nd fa-2x" aria-hidden="true"></i>
       
+    </a>
+  </p>
 
-    
 
-    
-    
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.js" integrity="sha256-EErZamuLefUnbMBQbsEqu1USa+btR2oIlCpBJbyD4/g=" crossorigin="anonymous"></script>
-    
 
-    
-    
-    <script>const code_highlighting = true;</script>
-    
 
-    
-    
-    <script>const isSiteThemeDark = false;</script>
-    
 
+  <p class="powered-by">
     
     
     
+      
+      
+      
+      
+      
+      
+      Published with <a href="https://hugoblox.com/?utm_campaign=poweredby" target="_blank" rel="noopener">Hugo Blox Builder</a> — the free, <a href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">open source</a> website builder that empowers creators.
     
-    
-    
-    <script>
-      const search_config = {"indexURI":"/index.json","minLength":1,"threshold":0.3};
-      const i18n = {"no_results":"No results found","placeholder":"Search...","results":"results found"};
-      const content_type = {
-        'post': "Posts",
-        'project': "Projects",
-        'publication' : "Publications",
-        'talk' : "Talks"
-        };
-    </script>
-    
+  </p>
+</footer>
 
+    </div>
     
-    
+  </div>
 
-    
-    
-    <script id="search-hit-fuse-template" type="text/x-template">
-      <div class="search-hit" id="summary-{{key}}">
+  
+
+
+<script src="/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js"></script>
+
+
+
+
+  
+
+  
+  
+
+  
+
+
+
+
+
+
+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
+
+
+
+
+
+
+
+
+  
+  <script id="search-hit-fuse-template" type="text/x-template">
+    <div class="search-hit" id="summary-{{key}}">
       <div class="search-hit-content">
         <div class="search-hit-name">
           <a href="{{relpermalink}}">{{title}}</a>
@@ -630,42 +901,18 @@ <h1>Categories</h1>
           <p class="search-hit-description">{{snippet}}</p>
         </div>
       </div>
-      </div>
-    </script>
-    
+    </div>
+  </script>
+  
+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
+    <script src="https://cdn.jsdelivr.net/gh/julmot/mark.js@8.11.1/dist/jquery.mark.min.js" integrity="sha512-mhbv5DqBMgrWL+32MmsDOt/OAvqr/cHimk6B8y/bx/xS88MVkYGPiVv2ixKVrkywF2qHplNRUvFsAHUdxZ3Krg==" crossorigin="anonymous"></script>
+  
 
-    
-    
-    <script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/3.2.1/fuse.min.js" integrity="sha256-VzgmKYmhsGNNN4Ph1kMW+BjoYJM2jV5i4IlFoeZA9XI=" crossorigin="anonymous"></script>
-    <script src="https://cdnjs.cloudflare.com/ajax/libs/mark.js/8.11.1/jquery.mark.min.js" integrity="sha256-4HLtjeVgH0eIB3aZ9mLYF6E8oU5chNdjU6p6rrXpl9U=" crossorigin="anonymous"></script>
-    
 
-    
-    
 
-    
-    
 
-    
-    
 
-    
-    
-    
-    
-    
-    
-    
-    
-    
-      
-    
-    
-    
-    
-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
-    
 
 
 
@@ -674,31 +921,53 @@ <h1>Categories</h1>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
-    <a href="https://gohugo.io" target="_blank" rel="noopener">Hugo</a>.
 
-    
-    <span class="float-right" aria-hidden="true">
-      <a href="#" class="back-to-top">
-        <span class="button_icon">
-          <i class="fas fa-chevron-up fa-2x"></i>
-        </span>
-      </a>
-    </span>
-    
-  </p>
-</footer>
 
-  </div>
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
   
+  
+
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
 
   
 <div id="modal" class="modal fade" role="dialog">
@@ -711,7 +980,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -726,5 +996,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 </body>
 </html>
diff --git a/categories/index.xml b/categories/index.xml
index fd21435..f8aca86 100644
--- a/categories/index.xml
+++ b/categories/index.xml
@@ -5,9 +5,9 @@
     <link>https://frankschae.github.io/categories/</link>
       <atom:link href="https://frankschae.github.io/categories/index.xml" rel="self" type="application/rss+xml" />
     <description>Categories</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
       <title>Categories</title>
       <link>https://frankschae.github.io/categories/</link>
     </image>
diff --git a/categories/page/1/index.html b/categories/page/1/index.html
index 5c469d1..d7e0edc 100644
--- a/categories/page/1/index.html
+++ b/categories/page/1/index.html
@@ -1 +1,10 @@
-<!DOCTYPE html><html><head><title>https://frankschae.github.io/categories/</title><link rel="canonical" href="https://frankschae.github.io/categories/"/><meta name="robots" content="noindex"><meta charset="utf-8" /><meta http-equiv="refresh" content="0; url=https://frankschae.github.io/categories/" /></head></html>
\ No newline at end of file
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/categories/</title>
+    <link rel="canonical" href="https://frankschae.github.io/categories/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/categories/">
+  </head>
+</html>
diff --git a/css/libs/chroma/dracula.min.css b/css/libs/chroma/dracula.min.css
new file mode 100644
index 0000000..d31f6fc
--- /dev/null
+++ b/css/libs/chroma/dracula.min.css
@@ -0,0 +1 @@
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diff --git a/css/libs/chroma/github-light.min.css b/css/libs/chroma/github-light.min.css
new file mode 100644
index 0000000..dc5acf8
--- /dev/null
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new file mode 100644
index 0000000..35a4a83
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+          
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+              <span>Automatic</span>
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+
+      </ul>
+
+    </div>
+  </nav>
+</header>
+
+
+  </div>
+
+  <div class="page-body">
+    
+    
+    
+
+    
+
+
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+
+
+
+
+
+
+
+
+
+
+  
+
+  
+  
+  
+    
+  
+<div class="universal-wrapper pt-3">
+  <h1>Recent &amp; Upcoming Talks</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  <div class="row">
+    <div class="col-lg-12">
+
+      
+
+      
+      <div class="row" id="talk_list">
+        <div class="col-lg-2">
+          <h3>2030</h3>
+        </div>
+        <div class="col-lg-10">
+
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+
+
+
+
+
+  
+
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+    <div class="section-subheading article-title mb-0 mt-0">
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+    </div>
+
+    
+    <a href="/talk/example-talk/"  class="summary-link">
+      <div class="article-style">
+        An example talk using Hugo Blox Builder&rsquo;s Markdown slides feature.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+      <div>
+        <span>
+          Jun 1, 2030 1:00 PM &mdash; 3:00 PM
+        </span>
+        
+        <span class="middot-divider"></span>
+        <span>Hugo Blox Builder HQ</span>
+        
+      </div>
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+
+
+
+
+  
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+    Slides
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+  
+
+
+
+
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://twitter.com/georgecushen" target="_blank" rel="noopener">
+    <i class="fab fa-twitter mr-1"></i>Follow</a>
+
+
+    </div>
+    
+
+  </div>
+  <div class="ml-3">
+    
+    
+      
+      
+      <a href="/talk/example-talk/" >
+        <img src="/talk/example-talk/featured_hu3d03a01dcc18bc5be0e67db3d8d209a6_620088_dfa11c4d594aa74ac0e82069e2706b64.webp" height="107" width="150"
+            alt="Example Talk" loading="lazy">
+      </a>
+    
+  </div>
+</div>
+
+          
+
+        </div>
+      </div>
+      
+
+    </div>
+  </div>
+
+</div>
+  </div>
+
+  <div class="page-footer">
+    
+    
+    <div class="container">
+      <footer class="site-footer">
+
+  
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+  
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+
+  
+  
+    
+  
+  
+    
+  
+
+  
+
+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
+  
+
+  <p class="powered-by footer-license-icons">
+    <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank" aria-label="Creative Commons">
+      <i class="fab fa-creative-commons fa-2x" aria-hidden="true"></i>
+      <i class="fab fa-creative-commons-by fa-2x" aria-hidden="true"></i>
+      
+        <i class="fab fa-creative-commons-nc fa-2x" aria-hidden="true"></i>
+      
+      
+        <i class="fab fa-creative-commons-nd fa-2x" aria-hidden="true"></i>
+      
+    </a>
+  </p>
+
+
+
+
+
+  <p class="powered-by">
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+      <div class="modal-header">
+        <h5 class="modal-title">Cite</h5>
+        <button type="button" class="close" data-dismiss="modal" aria-label="Close">
+          <span aria-hidden="true">&times;</span>
+        </button>
+      </div>
+      <div class="modal-body">
+        
+        <pre><code></code></pre>
+      </div>
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new file mode 100644
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+++ b/event/index.xml
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+<?xml version="1.0" encoding="utf-8" standalone="yes" ?>
+<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
+  <channel>
+    <title>Recent &amp; Upcoming Talks | FS</title>
+    <link>https://frankschae.github.io/event/</link>
+      <atom:link href="https://frankschae.github.io/event/index.xml" rel="self" type="application/rss+xml" />
+    <description>Recent &amp; Upcoming Talks</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sat, 01 Jun 2030 13:00:00 +0000</lastBuildDate>
+    <image>
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+      <title>Recent &amp; Upcoming Talks</title>
+      <link>https://frankschae.github.io/event/</link>
+    </image>
+    
+    <item>
+      <title>Example Talk</title>
+      <link>https://frankschae.github.io/talk/example-talk/</link>
+      <pubDate>Sat, 01 Jun 2030 13:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/talk/example-talk/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Click on the &lt;strong&gt;Slides&lt;/strong&gt; button above to view the built-in slides feature.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Slides can be added in a few ways:&lt;/p&gt;
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+&lt;li&gt;&lt;strong&gt;Create&lt;/strong&gt; slides using Hugo Blox Builder&amp;rsquo;s &lt;a href=&#34;https://docs.hugoblox.com/reference/content-types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;em&gt;Slides&lt;/em&gt;&lt;/a&gt; feature and link using &lt;code&gt;slides&lt;/code&gt; parameter in the front matter of the talk file&lt;/li&gt;
+&lt;li&gt;&lt;strong&gt;Upload&lt;/strong&gt; an existing slide deck to &lt;code&gt;static/&lt;/code&gt; and link using &lt;code&gt;url_slides&lt;/code&gt; parameter in the front matter of the talk file&lt;/li&gt;
+&lt;li&gt;&lt;strong&gt;Embed&lt;/strong&gt; your slides (e.g. Google Slides) or presentation video on this page using &lt;a href=&#34;https://docs.hugoblox.com/reference/markdown/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shortcodes&lt;/a&gt;.&lt;/li&gt;
+&lt;/ul&gt;
+&lt;p&gt;Further event details, including &lt;a href=&#34;https://docs.hugoblox.com/reference/markdown/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;page elements&lt;/a&gt; such as image galleries, can be added to the body of this page.&lt;/p&gt;
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+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
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+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
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+  
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+  
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+    
+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
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+  <meta name="author" content="Frank Schäfer" />
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+
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+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/home/" />
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+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
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+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
+<meta property="twitter:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" />
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+<meta property="og:type" content="website" />
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+  
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+  
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+  <title>Homes | FS</title>
+
+  
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+
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+
+
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
+
+  
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+  
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+
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+  <div class="container">
+    <section class="search-header">
+
+      <div class="row no-gutters justify-content-between mb-3">
+        <div class="col-6">
+          <h1>Search</h1>
+        </div>
+        <div class="col-6 col-search-close">
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+        </div>
+      </div>
+
+      <div id="search-box">
+        
+        <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
+        
+      </div>
+
+      
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+
+      
+
+    </section>
+    <section class="section-search-results">
+
+      <div id="search-hits">
+        
+      </div>
+
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+  </div>
+</aside>
+
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+  <div class="page-header header--fixed">
+  
+  
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+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
+
+      
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+        
+        <ul class="navbar-nav d-md-inline-flex">
+          
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+          
+
+          
+            
+              
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+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
+
+          
+
+          
+          
+          
+            
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+          
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+          
+          
+          
+          
+
+          
+            
+              
+              
+            
+            
+              
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+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
+
+          
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+          
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+
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+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
+
+          
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+
+          
+
+          
+          
+          
+          
+
+          
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+                
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+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
+          
+          
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+
+          
+
+          
+          
+          
+          
+
+          
+            
+              
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+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
+
+        
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
+        
+        
+          
+        
+
+        
+        
+        
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
+        
+
+        
+        
+        
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
+            </a>
+          </div>
+        </li>
+        
+
+        
+        
+
+      </ul>
+
+    </div>
+  </nav>
+</header>
+
+
+  </div>
+
+  <div class="page-body">
+    
+    
+    
+
+    
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+
+  
+  
+  
+    
+  
+<div class="universal-wrapper pt-3">
+  <h1>Homes</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  
+
+
+  
+
+  
+  
+
+  
+
+  
+
+</div>
+  </div>
+
+  <div class="page-footer">
+    
+    
+    <div class="container">
+      <footer class="site-footer">
+
+  
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+  
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+
+  
+  
+    
+  
+  
+    
+  
+
+  
+
+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
+  
+
+  <p class="powered-by footer-license-icons">
+    <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank" aria-label="Creative Commons">
+      <i class="fab fa-creative-commons fa-2x" aria-hidden="true"></i>
+      <i class="fab fa-creative-commons-by fa-2x" aria-hidden="true"></i>
+      
+        <i class="fab fa-creative-commons-nc fa-2x" aria-hidden="true"></i>
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+      
+        <i class="fab fa-creative-commons-nd fa-2x" aria-hidden="true"></i>
+      
+    </a>
+  </p>
+
+
+
+
+
+  <p class="powered-by">
+    
+    
+    
+      
+      
+      
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+      
+      
+      Published with <a href="https://hugoblox.com/?utm_campaign=poweredby" target="_blank" rel="noopener">Hugo Blox Builder</a> — the free, <a href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">open source</a> website builder that empowers creators.
+    
+  </p>
+</footer>
+
+    </div>
+    
+  </div>
+
+  
+
+
+<script src="/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js"></script>
+
+
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+  
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+  
+  
+
+  
+
+
+
+
+
+
+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
+
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+
+  
+  <script id="search-hit-fuse-template" type="text/x-template">
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+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
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+  
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+  
+  
+  
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+  
+  
+
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
+
+  
+<div id="modal" class="modal fade" role="dialog">
+  <div class="modal-dialog">
+    <div class="modal-content">
+      <div class="modal-header">
+        <h5 class="modal-title">Cite</h5>
+        <button type="button" class="close" data-dismiss="modal" aria-label="Close">
+          <span aria-hidden="true">&times;</span>
+        </button>
+      </div>
+      <div class="modal-body">
+        
+        <pre><code></code></pre>
+      </div>
+      <div class="modal-footer">
+        <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
+          <i class="fas fa-copy"></i> Copy
+        </a>
+        <a class="btn btn-outline-primary my-1 js-download-cite" href="#" target="_blank">
+          <i class="fas fa-download"></i> Download
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+        <div id="modal-error"></div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
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+
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+
+</body>
+</html>
diff --git a/home/index.xml b/home/index.xml
new file mode 100644
index 0000000..e2a8d17
--- /dev/null
+++ b/home/index.xml
@@ -0,0 +1,16 @@
+<?xml version="1.0" encoding="utf-8" standalone="yes" ?>
+<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
+  <channel>
+    <title>Homes | FS</title>
+    <link>https://frankschae.github.io/home/</link>
+      <atom:link href="https://frankschae.github.io/home/index.xml" rel="self" type="application/rss+xml" />
+    <description>Homes</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language>
+    <image>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Homes</title>
+      <link>https://frankschae.github.io/home/</link>
+    </image>
+    
+  </channel>
+</rss>
diff --git a/home/page/1/index.html b/home/page/1/index.html
new file mode 100644
index 0000000..08c09a1
--- /dev/null
+++ b/home/page/1/index.html
@@ -0,0 +1,10 @@
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/home/</title>
+    <link rel="canonical" href="https://frankschae.github.io/home/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/home/">
+  </head>
+</html>
diff --git a/index.html b/index.html
index f98fbc2..305136e 100644
--- a/index.html
+++ b/index.html
@@ -1,78 +1,113 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/">
+
+
+
+
+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
-  <script src="/js/mathjax-config.js"></script>
   
 
   
+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
+
   
   
   
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/academicons/1.8.6/css/academicons.min.css" integrity="sha256-uFVgMKfistnJAfoCUQigIl+JfUaP47GrRKjf6CTPVmw=" crossorigin="anonymous">
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+    
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1.9.4/css/academicons.min.css" integrity="sha512-IW0nhlW5MgNydsXJO40En2EoCkTTjZhI3yuODrZIc8cQ4h1XcF53PsqDHa09NqnkXuIe0Oiyyj171BqZFwISBw==" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
     
     
     
+    
+      
       
     
     
       
+        
+      
+        
+      
+        
+      
+        
       
         
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/github.min.css" crossorigin="anonymous" title="hl-light">
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/dracula.min.css" crossorigin="anonymous" title="hl-dark" disabled>
+      
         
       
+    
     
 
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.css" integrity="sha256-SHMGCYmST46SoyGgo4YR/9AlK1vf3ff84Aq9yK4hdqM=" crossorigin="anonymous">
+    
+    
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
     
 
     
@@ -80,12 +115,6 @@
     
     
       
-
-      
-      
-
-      
-    
       
 
       
@@ -94,6 +123,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +131,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +139,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +147,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +155,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +163,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +171,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +179,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +187,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,67 +220,141 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
+  
+  
+  
+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
+
+  
+  
   
 
   
   
   
   
-  <link rel="stylesheet" href="/css/academic.css">
+  
+  
+  
+    
+    
+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
+  
 
   
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 
 
 
 
-  
 
 
-  
-  
-  <script src="https://identity.netlify.com/v1/netlify-identity-widget.js"></script>
-  
 
-  
-  <link rel="alternate" href="/index.xml" type="application/rss+xml" title="FS">
-  
 
-  <link rel="manifest" href="/index.webmanifest">
-  <link rel="icon" type="image/png" href="/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_2.png">
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-  <link rel="canonical" href="https://frankschae.github.io/">
 
   
   
   
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
+
+
+
+
+
   
+
+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
+
+
+
+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/" />
+<link rel="canonical" href="https://frankschae.github.io/" />
+
+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
+
+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
+<link rel="apple-touch-icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_180x180_fill_lanczos_center_3.png" />
+
+<meta name="theme-color" content="#1565c0" />
+
+
+
+
+
+
+
+
+
+
   
   
+
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+
+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
+<meta property="twitter:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" />
+
+
+
   
     
-    
-  
+    <meta property="profile:first_name" content="Frank">
+    <meta property="profile:last_name" content="Schäfer">
   
-  <meta property="twitter:card" content="summary">
-  
-  <meta property="og:site_name" content="FS">
-  <meta property="og:url" content="https://frankschae.github.io/">
-  <meta property="og:title" content="FS">
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+
+<meta property="og:type" content="profile" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/" />
+<meta property="og:title" content="FS" />
+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
+
   
-    <meta property="og:updated_time" content="2024-01-15T10:49:24-05:00">
+    <meta property="og:updated_time" content="2022-10-24T00:00:00&#43;00:00" />
   
 
-  
+
+
+
 
 <script type="application/ld+json">
 {
@@ -267,22 +373,58 @@
   
 
 
+
+
+  
+  
+  
+
+  
+  
+    <link rel="alternate" href="/index.xml" type="application/rss+xml" title="FS" />
+  
+
   
 
 
   
+  <title>FS</title>
+
+  
+  
+  
+  
+
+
+
+
+
 
 
 
 
 
-  <title>FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#navbar-main" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#navbar-main" class="page-wrapper   " data-wc-page-id="3976528693a0108357f4928017600865" >
+
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
+
+  
+
+
+<aside class="search-modal" id="search">
   <div class="container">
     <section class="search-header">
 
@@ -291,17 +433,23 @@
           <h1>Search</h1>
         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -314,6 +462,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
+  
   
 
 
@@ -322,153 +476,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    <button type="button" class="navbar-toggler" data-toggle="collapse"
-            data-target="#navbar-content" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
-    <span><i class="fas fa-bars"></i></span>
-    </button>
-    
 
-    
-    <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    
-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
-        
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
 
-        
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
-        
-        
+        <ul class="navbar-nav d-md-inline-flex">
+          
+
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about" data-target="#about"><span>Home</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#about" data-target="#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts" data-target="#posts"><span>Posts</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts" data-target="#posts"><span>Posts</span></a>
+          </li>
 
-        
-        
-        
           
-        
+          
 
-        
-        
-        
-        
-        
-        
+          
+
+          
+          
+          
+            
+          
+
+          
+
+          
           
           
           
+
+          
+            
+              
+              
+            
             
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects" data-target="#projects"><span>Research projects</span></a>
+          </li>
+
           
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#projects" data-target="#projects"><span>Research projects</span></a>
-        </li>
+          
 
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+            <a class="nav-link " href="/#software" data-target="#software"><span>Open source software</span></a>
+          </li>
+
           
           
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+            <a class="nav-link " href="/#contact" data-target="#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
 
         
+
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+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
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+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
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+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
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+              <span>Light</span>
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+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
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+  </div>
 
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+  <div class="page-body">
+    
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+    
 
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+    
 
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+<span class="js-widget-page d-none"></span>
 
 
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+<section id="about" class="home-section wg-about-biography  "  >
+ <div class="home-section-bg " >
+   
+ </div>
+  <div class="container">
+
   
 
-  <section id="about" class="home-section wg-about   "  >
-    <div class="container">
-      
+    
+
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+
 
 
 
@@ -615,11 +864,15 @@ <h1>Search</h1>
 
       
       
-      <img class="avatar avatar-circle" src="/author/frank-schafer/avatar_hu5bdd272e648badb42a595cb37d3ba435_2457515_270x270_fill_lanczos_center_2.png" alt="Frank Schäfer">
+      <img class="avatar avatar-circle"
+           width="270" height="270"
+           src="/authors/admin/avatar_hu5bdd272e648badb42a595cb37d3ba435_2457515_270x270_fill_lanczos_center_3.png" alt="Frank Schäfer">
       
 
       <div class="portrait-title">
+
         <h2>Frank Schäfer</h2>
+
         <h3>Postdoctoral researcher</h3>
 
         
@@ -645,7 +898,7 @@ <h3>
           
         
         <li>
-          <a href="/#contact" >
+          <a href="/#contact"  aria-label="envelope">
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@@ -662,7 +915,7 @@ <h3>
           
         
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-          <a href="https://twitter.com/_Frank_Schaefer" target="_blank" rel="noopener">
+          <a href="https://twitter.com/_Frank_Schaefer" target="_blank" rel="noopener" aria-label="twitter">
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-          <a href="https://scholar.google.com/citations?user=goAokcEAAAAJ&amp;hl=de&amp;oi=sra#" target="_blank" rel="noopener">
+          <a href="https://scholar.google.com/citations?user=goAokcEAAAAJ&amp;hl=de&amp;oi=sra#" target="_blank" rel="noopener" aria-label="google-scholar">
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         <li>
-          <a href="https://github.com/frankschae" target="_blank" rel="noopener">
+          <a href="https://github.com/frankschae" target="_blank" rel="noopener" aria-label="github">
             <i class="fab fa-github big-icon"></i>
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-          <a href="https://orcid.org/0000-0003-2684-4984" target="_blank" rel="noopener">
+          <a href="https://orcid.org/0000-0003-2684-4984" target="_blank" rel="noopener" aria-label="orcid">
             <i class="ai ai-orcid big-icon"></i>
           </a>
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@@ -723,128 +976,505 @@ <h3>
     
     <h1>Biography</h1>
 
-    <p>I am a postdoc in the 
-<a href="https://julia.mit.edu/" target="_blank" rel="noopener">Julia Lab</a> located in the Computer Science and Artificial Intelligence Laboratory (CSAIL) at the Massachusetts Institute of Technology (MIT). My research is focused on 
-<a href="https://sciml.ai/" target="_blank" rel="noopener">Scientific Machine Learning (SciML)</a>. I completed my PhD in physics in the 
-<a href="https://quantumtheory-bruder.physik.unibas.ch/en/" target="_blank" rel="noopener">Bruder group</a> within the &ldquo;Quantum Computing and Quantum Technology&rdquo; PhD school at the University of Basel. During my PhD, I participated in the Google Summer of Code (GSoC) 2020 and 2021 programs with the projects &ldquo;
-<a href="https://summerofcode.withgoogle.com/archive/2020/projects/5076877036748800/" target="_blank" rel="noopener">High weak order stochastic differential equation solvers and their utility in neural stochastic differential equations</a>&rdquo; within the Julia Language organization and &ldquo;
-<a href="https://summerofcode.withgoogle.com/projects/#5357798591823872" target="_blank" rel="noopener">Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis</a>&rdquo;  within the NumFocus organization, supervised by 
-<a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, 
-<a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, 
-<a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a>, and 
-<a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a>. Since 2020, I am a member of the 
-<a href="https://sciml.ai/" target="_blank" rel="noopener">SciML</a> open source software organization for scientific machine learning.</p>
+    
+    <div class="article-style">
+      <p>I am a postdoc in the <a href="https://julia.mit.edu/" target="_blank" rel="noopener">Julia Lab</a> located in the Computer Science and Artificial Intelligence Laboratory (CSAIL) at the Massachusetts Institute of Technology (MIT). My research is focused on <a href="https://sciml.ai/" target="_blank" rel="noopener">Scientific Machine Learning (SciML)</a>. I completed my PhD in physics in the <a href="https://quantumtheory-bruder.physik.unibas.ch/en/" target="_blank" rel="noopener">Bruder group</a> within the &ldquo;Quantum Computing and Quantum Technology&rdquo; PhD school at the University of Basel. During my PhD, I participated in the Google Summer of Code (GSoC) 2020 and 2021 programs with the projects &ldquo;<a href="https://summerofcode.withgoogle.com/archive/2020/projects/5076877036748800/" target="_blank" rel="noopener">High weak order stochastic differential equation solvers and their utility in neural stochastic differential equations</a>&rdquo; within the Julia Language organization and &ldquo;<a href="https://summerofcode.withgoogle.com/projects/#5357798591823872" target="_blank" rel="noopener">Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis</a>&rdquo;  within the NumFocus organization, supervised by <a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, <a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, <a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a>, and <a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a>. Since 2020, I am a member of the <a href="https://sciml.ai/" target="_blank" rel="noopener">SciML</a> open source software organization for scientific machine learning.</p>
+
+    </div>
+    
+
+    <div class="row">
+
+      
+      <div class="col-md-5">
+        <div class="section-subheading">Interests</div>
+        <ul class="ul-interests fa-ul mb-0">
+          
+          <li>
+            <i class="fa-li fa-solid fa-book-open"></i>
+            Probabilistic machine learning
+          </li>
+          
+          <li>
+            <i class="fa-li fa-solid fa-book-open"></i>
+            Differentiable &amp; probabilistic programming
+          </li>
+          
+          <li>
+            <i class="fa-li fa-solid fa-book-open"></i>
+            Quantum &amp; stochastic optimal control
+          </li>
+          
+          <li>
+            <i class="fa-li fa-solid fa-book-open"></i>
+            Parameter inference / estimation theory
+          </li>
+          
+          <li>
+            <i class="fa-li fa-solid fa-book-open"></i>
+            Stochastic processes and many-body physics
+          </li>
+          
+        </ul>
+      </div>
+      
+
+      
+      <div class="col-md-7">
+        <div class="section-subheading">Education</div>
+        <ul class="ul-edu fa-ul mb-0">
+          
+          <li>
+            <i class="fa-li fa-solid fa-graduation-cap"></i>
+            <div class="description">
+              <p class="course">PhD in Physics under the supervision of Prof. Dr. Christoph Bruder, 2022</p>
+              <p class="institution">Department of Physics, University of Basel</p>
+            </div>
+          </li>
+          
+          <li>
+            <i class="fa-li fa-solid fa-graduation-cap"></i>
+            <div class="description">
+              <p class="course">MSc in Physics under the supervision of Prof. Dr. Andreas Buchleitner, 2018</p>
+              <p class="institution">Department of Physics, Albert-Ludwigs-Universität Freiburg</p>
+            </div>
+          </li>
+          
+          <li>
+            <i class="fa-li fa-solid fa-graduation-cap"></i>
+            <div class="description">
+              <p class="course">BSc in Physics under the supervision of PD Dr. Thomas Wellens, 2015</p>
+              <p class="institution">Department of Physics, Albert-Ludwigs-Universität Freiburg</p>
+            </div>
+          </li>
+          
+        </ul>
+      </div>
+      
+
+    </div>
+  </div>
+</div>
+
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+  
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+  </div>
+</section>
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+
+
+<section id="posts" class="home-section wg-collection  "  >
+ <div class="home-section-bg " >
+   
+ </div>
+  <div class="container">
+
+  
+    <div class="row  ">
+    
+      
+        <div class="section-heading col-12 col-lg-4 mb-3 mb-lg-0 d-flex flex-column align-items-center align-items-lg-start">
+          <h1 class="mb-0">Posts</h1>
+          
+        </div>
+      
+    
+  
+
+    
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+
 
 
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+  
+
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/gsoc-2021/" >Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis</a>
+    </div>
+
+    
+    <a href="/post/gsoc-2021/"  class="summary-link">
+      <div class="article-style">
+        Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+        
+
+
+<div class="article-metadata">
+
+  
+  
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+  
+  <span class="article-date">
+    
+    
+      
+    
+    Aug 13, 2021
+  </span>
+  
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+  
+
+  
+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    3 min read
+  </span>
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+</div>
+
+      
+    </div>
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/abstract_differentiation/" >AbstractDifferentiation.jl for AD-backend agnostic code </a>
+    </div>
+
+    
+    <a href="/post/abstract_differentiation/"  class="summary-link">
+      <div class="article-style">
+        Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris&rsquo; article3 for a detailed examination.
+      </div>
+    </a>
+    
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+    <div class="stream-meta article-metadata">
+
+      
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+      
+        
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+<div class="article-metadata">
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+  <span class="article-date">
+    
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+      
+    
+    Aug 1, 2021
+  </span>
+  
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+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    11 min read
+  </span>
+  
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+</div>
+
+      
+    </div>
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+    
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+  </div>
+  <div class="ml-3">
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a>
+    </div>
+
+    
+    <a href="/post/bouncing_ball/"  class="summary-link">
+      <div class="article-style">
+        In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
 
       
-      <div class="col-md-5">
-        <h3>Interests</h3>
-        <ul class="ul-interests">
-          
-          <li>Machine learning &amp; data-driven approaches</li>
-          
-          <li>Differentiable &amp; probabilistic programming</li>
-          
-          <li>Sensitivity analysis</li>
-          
-          <li>Quantum &amp; stochastic optimal control</li>
-          
-          <li>Parameter inference / estimation theory</li>
-          
-          <li>Stochastic processes</li>
-          
-          <li>Quantum optics</li>
-          
-          <li>Many-body physics</li>
-          
-        </ul>
-      </div>
-      
 
       
-      <div class="col-md-7">
-        <h3>Education</h3>
-        <ul class="ul-edu fa-ul">
-          
-          <li>
-            <i class="fa-li fas fa-graduation-cap"></i>
-            <div class="description">
-              <p class="course">PhD in Physics under the supervision of Prof. Dr. Christoph Bruder, 2022</p>
-              <p class="institution">Department of Physics, University of Basel</p>
-            </div>
-          </li>
-          
-          <li>
-            <i class="fa-li fas fa-graduation-cap"></i>
-            <div class="description">
-              <p class="course">MSc in Physics under the supervision of Prof. Dr. Andreas Buchleitner, 2018</p>
-              <p class="institution">Department of Physics, Albert-Ludwigs-Universität Freiburg</p>
-            </div>
-          </li>
-          
-          <li>
-            <i class="fa-li fas fa-graduation-cap"></i>
-            <div class="description">
-              <p class="course">BSc in Physics under the supervision of PD Dr. Thomas Wellens, 2015</p>
-              <p class="institution">Department of Physics, Albert-Ludwigs-Universität Freiburg</p>
-            </div>
-          </li>
-          
-        </ul>
-      </div>
-      
+        
 
-    </div>
-  </div>
-</div>
 
-    </div>
-  </section>
+<div class="article-metadata">
 
   
   
   
-
   
+  <div>
+    
 
+  <span >
+      Frank Schäfer and Moritz Schauer</span>
+  </div>
   
-
   
 
   
-
+  <span class="article-date">
+    
+    
+      
+    
+    Jul 16, 2021
+  </span>
   
 
   
-  
 
   
-  
+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    19 min read
+  </span>
   
 
   
   
-  
   
   
 
   
   
 
-  <section id="posts" class="home-section wg-pages   "  >
-    <div class="container">
-      
-
-
-
-
+</div>
 
+      
+    </div>
 
+    
 
+  </div>
+  <div class="ml-3">
+    
+    
+  </div>
+</div>
 
   
+    
 
 
 
@@ -858,38 +1488,81 @@ <h3>Education</h3>
 
 
 
+  
 
 
+<div class="media stream-item view-compact">
+  <div class="media-body">
 
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
+    </div>
 
+    
+    <a href="/post/shadowing/"  class="summary-link">
+      <div class="article-style">
+        In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
+      </div>
+    </a>
+    
 
+    <div class="stream-meta article-metadata">
 
+      
 
+      
+        
 
 
-
+<div class="article-metadata">
 
   
+  
 
+  
+  <span class="article-date">
+    
+    
+      
+    
+    Jul 2, 2021
+  </span>
+  
 
+  
 
+  
+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    9 min read
+  </span>
+  
 
+  
+  
+  
+  
 
   
+  
 
+</div>
+
+      
+    </div>
 
-<div class="row">
-  <div class="col-12 col-lg-4 section-heading">
-    <h1>Posts</h1>
     
-  </div>
-  <div class="col-12 col-lg-8">
 
+  </div>
+  <div class="ml-3">
+    
     
+  </div>
+</div>
 
+  
     
-      
-        
+
 
 
 
@@ -905,17 +1578,19 @@ <h1>Posts</h1>
   
 
 
-<div class="media stream-item">
+<div class="media stream-item view-compact">
   <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/post/gsoc-2021/" >Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/hybridde/" >Neural Hybrid Differential Equations</a>
+    </div>
 
     
-    <div class="article-style">
-      Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.
-    </div>
+    <a href="/post/hybridde/"  class="summary-link">
+      <div class="article-style">
+        I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
+      </div>
+    </a>
     
 
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-    Aug 13, 2021
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-    3 min read
+    7 min read
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+  
 
   
   
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   </div>
 </div>
 
+  
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+  
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+  
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+    
+    
       
     
+
+    
+    
+    
+    
       
-        
+    
+
+    
+
+    <div class="see-all">
+      <a href="/post/">
+        See all posts
+        <i class="fas fa-angle-right"></i>
+      </a>
+    </div>
+  
+
+</div>
+
+
+  
+    </div>
+  
+
+  </div>
+</section>
+
+  
+
+  
+  
+    
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+
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+
 
 
 
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+
+<section id="projects" class="home-section wg-collection  "  >
+ <div class="home-section-bg " >
+   
+ </div>
+  <div class="container">
+
+  
+    <div class="row  ">
+    
+      
+        <div class="section-heading col-12 col-lg-4 mb-3 mb-lg-0 d-flex flex-column align-items-center align-items-lg-start">
+          <h1 class="mb-0">Research projects</h1>
+          
+        </div>
+      
+    
+  
+
+    
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+
   
 
 
-<div class="media stream-item">
-  <div class="media-body">
-
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/post/abstract_differentiation/" >AbstractDifferentiation.jl for AD-backend agnostic code </a>
-    </h3>
 
-    
-    <div class="article-style">
-      Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris' article3 for a detailed examination.
-    </div>
-    
 
-    <div class="stream-meta article-metadata">
 
-      
 
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-        
 
 
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-  <span class="article-date">
-    
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-    
-    Aug 1, 2021
-  </span>
   
-
   
 
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-  <span class="article-reading-time">
-    11 min read
-  </span>
-  
 
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-      
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-  <div class="ml-3">
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-      
-    
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-        
+
 
 
 
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-<div class="media stream-item">
-  <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a>
-    </h3>
 
-    
-    <div class="article-style">
-      In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
-    </div>
+<div class="col-12 col-lg-8">
+
+  
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+  
     
 
-    <div class="stream-meta article-metadata">
 
-      
 
-      
-        
 
 
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-  <div>
-    
 
   
-  <span>Frank Schäfer and Moritz Schauer</span>
-  </div>
-  
-  
 
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-  <span class="article-date">
-    
-    
-      
-    
-    Jul 16, 2021
-  </span>
-  
 
-  
 
-  
-  <span class="middot-divider"></span>
-  <span class="article-reading-time">
-    19 min read
-  </span>
-  
 
-  
-  
-  
 
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+
+
   
 
-</div>
 
-      
-    </div>
+<div class="media stream-item view-compact">
+  <div class="media-body">
 
-    
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/stochasticad/" >Automatic Differentiation of Programs with Discrete Randomness</a>
+    </div>
 
-  </div>
-  <div class="ml-3">
     
+    <a href="/project/stochasticad/"  class="summary-link">
+      <div class="article-style">
+        We develop and implement AD algorithms for handling programs that can contain discrete randomness.
+      </div>
+    </a>
     
-  </div>
-</div>
+
+    <div class="stream-meta article-metadata">
 
       
-    
+
       
+      <div>
         
 
+  <span >
+      FS in Collaboration With Gaurav Arya</span>, <span >
+      Moritz Schauer</span>, <span >
+      Ruben Seyer</span>, <span >
+      Alex Lew</span>, <span >
+      Mathieu Huot</span>, <span >
+      Kartik Chandra</span>, <span >
+      Vikash Mansinghka</span>, <span >
+      Jonathan Ragan-Kelley</span>, <span >
+      Chris Rackauckas</span>
+      </div>
+      
+    </div>
 
+    
+    <div class="btn-links">
+      
 
 
 
@@ -1165,81 +1893,81 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
+  
 
 
-  
 
 
-<div class="media stream-item">
-  <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
-    </h3>
 
+  
+  
     
-    <div class="article-style">
-      In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
-    </div>
-    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/gaurav-arya/StochasticAD.jl" target="_blank" rel="noopener">
+  Code
+</a>
+
+
+
+
+
+
+
 
-    <div class="stream-meta article-metadata">
 
-      
 
-      
-        
 
 
-<div class="article-metadata">
 
   
   
-
   
-  <span class="article-date">
-    
     
-      
+  
+  
+  
+  
+  
     
-    Jul 2, 2021
-  </span>
   
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2210.08572" target="_blank" rel="noopener">
+    NeurIPS 2022</a>
 
   
-
   
-  <span class="middot-divider"></span>
-  <span class="article-reading-time">
-    9 min read
-  </span>
   
-
+    
   
   
   
-
   
   
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2306.07961" target="_blank" rel="noopener">
+    ICML 2023 Workshop</a>
 
-</div>
 
-      
     </div>
-
     
 
   </div>
   <div class="ml-3">
     
     
+      
+      
+      <a href="/project/stochasticad/" >
+        <img src="/project/stochasticad/featured_hud4f779b9fd6ed4129756e25755668fb0_498844_0c7aaabe77c3eece349770d347a2a364.webp" height="125" width="150"
+            alt="Automatic Differentiation of Programs with Discrete Randomness" loading="lazy">
+      </a>
+    
   </div>
 </div>
 
-      
+  
     
-      
-        
 
 
 
@@ -1247,6 +1975,9 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
+  
+
+
 
 
 
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-<div class="media stream-item">
+<div class="media stream-item view-compact">
   <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/post/hybridde/" >Neural Hybrid Differential Equations</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a>
+    </div>
 
     
-    <div class="article-style">
-      I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
-    </div>
+    <a href="/project/controlbounds/"  class="summary-link">
+      <div class="article-style">
+        We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -1273,143 +2006,96 @@ <h3 class="article-title mb-0 mt-0">
       
 
       
+      <div>
         
 
+  <span >
+      FS in Collaboration With Flemming Holtorf</span>, <span >
+      Julian Arnold</span>, <span >
+      Chris Rackauckas</span>, <span >
+      And Alan Edelman</span>
+      </div>
+      
+    </div>
 
-<div class="article-metadata">
-
-  
-  
-
-  
-  <span class="article-date">
-    
     
+    <div class="btn-links">
       
-    
-    Jun 16, 2021
-  </span>
-  
 
-  
 
-  
-  <span class="middot-divider"></span>
-  <span class="article-reading-time">
-    7 min read
-  </span>
-  
 
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-  
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+
+
   
 
-</div>
 
-      
-    </div>
 
-    
 
-  </div>
-  <div class="ml-3">
-    
-    
-  </div>
-</div>
 
-      
-    
 
-    
-    <div class="see-all">
-      <a href="/post/">
-        See all posts
-        <i class="fas fa-angle-right"></i>
-      </a>
-    </div>
-    
 
-  </div>
-</div>
 
-    </div>
-  </section>
 
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-  
-  
 
-  
 
-  
 
-  
 
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-  
 
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+
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+  
+  
+  
+    
+  
   
   
-
   
   
+    
   
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2304.03366" target="_blank" rel="noopener">
+    arXiv preprint</a>
 
   
   
   
+    
   
   
-
   
   
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://ieeexplore.ieee.org/abstract/document/10313626" target="_blank" rel="noopener">
+    QCE</a>
 
-  <section id="projects" class="home-section wg-portfolio   "  >
-    <div class="container">
-      
-
-
-
-
-
-
-
-
-
-
-<div class="row">
-  <div class="col-12 col-lg-4 section-heading">
 
-    <h1>Research projects</h1>
+    </div>
     
 
   </div>
-  <div class="col-12 col-lg-8">
-
-
-
+  <div class="ml-3">
     
-
     
-
-    <div class=" row js-layout-row ">
       
+      
+      <a href="/project/controlbounds/" >
+        <img src="/project/controlbounds/featured_hu9a224a6f0b76f74a72bd3d29668f06be_15378_efa414e40070ef54cd7713cdec8327e3.webp" height="91" width="150"
+            alt="Performance Bounds for Quantum Control" loading="lazy">
+      </a>
+    
+  </div>
+</div>
 
-        
-        
-        
-
-        
-          
-
-
-<div class="col-12 isotope-item ">
   
+    
 
 
 
@@ -1417,28 +2103,30 @@ <h1>Research projects</h1>
 
 
 
-
-  
   
 
 
 
 
 
+
+
   
 
 
-<div class="media stream-item">
+<div class="media stream-item view-compact">
   <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/project/stochasticad/" >Automatic Differentiation of Programs with Discrete Randomness</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a>
+    </div>
 
     
-    <div class="article-style">
-      We develop and implement AD algorithms for handling programs that can contain discrete randomness.
-    </div>
+    <a href="/project/dp_for_control/"  class="summary-link">
+      <div class="article-style">
+        Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -1449,8 +2137,12 @@ <h3 class="article-title mb-0 mt-0">
       <div>
         
 
-  
-  <span>FS in collaboration with Gaurav Arya</span>, <span>Moritz Schauer</span>, <span>Ruben Seyer</span>, <span>Alex Lew</span>, <span>Mathieu Huot</span>, <span>Kartik Chandra</span>, <span>Vikash Mansinghka</span>, <span>Jonathan Ragan-Kelley</span>, <span>Chris Rackauckas</span>
+  <span >
+      FS in Collaboration With Pavel Sekatski</span>, <span >
+      Martin Koppenhöfer</span>, <span >
+      Niels Lörch</span>, <span >
+      Christoph Bruder</span>, <span >
+      And Michal Kloc</span>
       </div>
       
     </div>
@@ -1477,7 +2169,7 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-<a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://github.com/gaurav-arya/StochasticAD.jl" target="_blank" rel="noopener">
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/frankschae/Control-of-Stochastic-Quantum-Dynamics-with-Differentiable-Programming" target="_blank" rel="noopener">
   Code
 </a>
 
@@ -1504,10 +2196,22 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://arxiv.org/abs/2210.08572" target="_blank" rel="noopener">
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
+    SDE control paper</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
     
-    NeurIPS 2022
-  </a>
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
+    ODE control paper</a>
 
   
   
@@ -1520,10 +2224,22 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://arxiv.org/abs/2306.07961" target="_blank" rel="noopener">
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=4" target="_blank" rel="noopener">
+    SIAM CSE 2021 talk</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
     
-    ICML 2023 Workshop
-  </a>
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=v8mJVGVdkNQ&amp;list=PLWIVj90xdDE-2eeyFuiooxWcF8kw323Iv&amp;index=3&amp;t=0s" target="_blank" rel="noopener">
+    CMD2020GEFES talk</a>
 
 
     </div>
@@ -1533,30 +2249,18 @@ <h3 class="article-title mb-0 mt-0">
   <div class="ml-3">
     
     
-    
-    <a href="/project/stochasticad/" >
-      <img src="/project/stochasticad/featured_hud4f779b9fd6ed4129756e25755668fb0_498844_150x0_resize_lanczos_2.png" alt="">
-    </a>
+      
+      
+      <a href="/project/dp_for_control/" >
+        <img src="/project/dp_for_control/featured_hudf85e1c26ad74a07a00aa9287e088b01_236898_6e426bbd03dcc15fb72201019eec9119.webp" height="113" width="150"
+            alt="Control of (Stochastic) Quantum Dynamics with Differentiable Programming" loading="lazy">
+      </a>
     
   </div>
 </div>
 
-</div>
-
-        
-
-      
-
-        
-        
-        
-
-        
-          
-
-
-<div class="col-12 isotope-item js-id-quantum-control js-id-convex-optimization js-id-stochastic-optimal-control js-id-sum-of-squares">
   
+    
 
 
 
@@ -1564,28 +2268,30 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
-
-  
   
 
 
 
 
 
+
+
   
 
 
-<div class="media stream-item">
+<div class="media stream-item view-compact">
   <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/reconstruction/" >Automated Data-driven Reconstruction of Physical Properties</a>
+    </div>
 
     
-    <div class="article-style">
-      We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
-    </div>
+    <a href="/project/reconstruction/"  class="summary-link">
+      <div class="article-style">
+        We address the challenges of connecting measurements of a physical system to an underlying theoretical description.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -1596,8 +2302,18 @@ <h3 class="article-title mb-0 mt-0">
       <div>
         
 
-  
-  <span>FS in collaboration with Flemming Holtorf</span>, <span>Julian Arnold</span>, <span>Chris Rackauckas</span>, <span>and Alan Edelman</span>
+  <span >
+      FS in Collaboration With Eliska Greplova</span>, <span >
+      Tomoki Hirosawa</span>, <span >
+      Jozef Bucko</span>, <span >
+      František Herman</span>, <span >
+      Rebekka Garreis</span>, <span >
+      Chuyao Tong</span>, <span >
+      Annika Kurzmann</span>, <span >
+      Thomas Ihn</span>, <span >
+      Hideaki Maebashi</span>, <span >
+      Hiroyasu Matsuura</span>, <span >
+      And Masao Ogata</span>
       </div>
       
     </div>
@@ -1643,10 +2359,8 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://arxiv.org/abs/2304.03366" target="_blank" rel="noopener">
-    
-    arXiv preprint
-  </a>
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.19.024015" target="_blank" rel="noopener">
+    Phys. Rev. Applied</a>
 
   
   
@@ -1659,10 +2373,8 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://ieeexplore.ieee.org/abstract/document/10313626" target="_blank" rel="noopener">
-    
-    QCE
-  </a>
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.jps.jp/doi/abs/10.7566/JPSJ.91.114603" target="_blank" rel="noopener">
+    J. Phys. Soc. Jpn.</a>
 
 
     </div>
@@ -1672,30 +2384,18 @@ <h3 class="article-title mb-0 mt-0">
   <div class="ml-3">
     
     
-    
-    <a href="/project/controlbounds/" >
-      <img src="/project/controlbounds/featured_hu9a224a6f0b76f74a72bd3d29668f06be_15378_150x0_resize_lanczos_2.png" alt="">
-    </a>
+      
+      
+      <a href="/project/reconstruction/" >
+        <img src="/project/reconstruction/featured_hub0c7e03472f12c4e5847301dd7a71967_835003_727e3f710288f4a2fac7811253811418.webp" height="134" width="150"
+            alt="Automated Data-driven Reconstruction of Physical Properties" loading="lazy">
+      </a>
     
   </div>
 </div>
 
-</div>
-
-        
-
-      
-
-        
-        
-        
-
-        
-          
-
-
-<div class="col-12 isotope-item js-id-SciML js-id-differentiable-programming js-id-NODE js-id-NSDE js-id-adjoint-sensitivity-methods js-id-automatic-differentiation js-id-quantum-control">
   
+    
 
 
 
@@ -1703,28 +2403,30 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
-
-  
   
 
 
 
 
 
+
+
   
 
 
-<div class="media stream-item">
+<div class="media stream-item view-compact">
   <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a>
+    </div>
 
     
-    <div class="article-style">
-      Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
-    </div>
+    <a href="/project/ml_for_pt/"  class="summary-link">
+      <div class="article-style">
+        Data-driven methods based on sample instances of the state of a physical system as a function of the system&rsquo;s parameters.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -1735,8 +2437,16 @@ <h3 class="article-title mb-0 mt-0">
       <div>
         
 
-  
-  <span>FS in collaboration with Pavel Sekatski</span>, <span>Martin Koppenhöfer</span>, <span>Niels Lörch</span>, <span>Christoph Bruder</span>, <span>and Michal Kloc</span>
+  <span >
+      FS in Collaboration With Julian Arnold</span>, <span >
+      Flemming Holtorf</span>, <span >
+      Eliska Greplova</span>, <span >
+      Agnes Valenti</span>, <span >
+      Martin Zonda</span>, <span >
+      Axel Lode</span>, <span >
+      Gregor Boschung</span>, <span >
+      Sebastian Huber</span>, <span >
+      And Niels Lörch</span>
       </div>
       
     </div>
@@ -1763,7 +2473,7 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-<a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://github.com/frankschae/Control-of-Stochastic-Quantum-Dynamics-with-Differentiable-Programming" target="_blank" rel="noopener">
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/arnoldjulian/Replacing-neural-networks-by-optimal-analytical-predictors-for-the-detection-of-phase-transitions" target="_blank" rel="noopener">
   Code
 </a>
 
@@ -1790,10 +2500,8 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
-    
-    SDE control paper
-  </a>
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
+    PRE</a>
 
   
   
@@ -1806,10 +2514,8 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
-    
-    ODE control paper
-  </a>
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/1367-2630/ab7771/meta" target="_blank" rel="noopener">
+    NJP</a>
 
   
   
@@ -1822,10 +2528,8 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=4" target="_blank" rel="noopener">
-    
-    SIAM CSE 2021 talk
-  </a>
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.033052" target="_blank" rel="noopener">
+    PRResearch</a>
 
   
   
@@ -1838,10 +2542,22 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://www.youtube.com/watch?v=v8mJVGVdkNQ&amp;list=PLWIVj90xdDE-2eeyFuiooxWcF8kw323Iv&amp;index=3&amp;t=0s" target="_blank" rel="noopener">
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.12.031044" target="_blank" rel="noopener">
+    PRX</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
     
-    CMD2020GEFES talk
-  </a>
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2311.09128" target="_blank" rel="noopener">
+    NeurIPS Workshop</a>
 
 
     </div>
@@ -1851,30 +2567,18 @@ <h3 class="article-title mb-0 mt-0">
   <div class="ml-3">
     
     
-    
-    <a href="/project/dp_for_control/" >
-      <img src="/project/dp_for_control/featured_hudf85e1c26ad74a07a00aa9287e088b01_236898_150x0_resize_q90_lanczos.jpeg" alt="">
-    </a>
+      
+      
+      <a href="/project/ml_for_pt/" >
+        <img src="/project/ml_for_pt/featured_hu3194460ba82e7df78553a9c87bbc0c72_237834_311c97c5d06b595a95a2178d807360ed.webp" height="150" width="150"
+            alt="Machine Learning for Phase Transitions" loading="lazy">
+      </a>
     
   </div>
 </div>
 
-</div>
-
-        
-
-      
-
-        
-        
-        
-
-        
-          
-
-
-<div class="col-12 isotope-item ">
   
+    
 
 
 
@@ -1882,28 +2586,30 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
-
-  
   
 
 
 
 
 
+
+
   
 
 
-<div class="media stream-item">
+<div class="media stream-item view-compact">
   <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/project/reconstruction/" >Automated Data-driven Reconstruction of Physical Properties</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a>
+    </div>
 
     
-    <div class="article-style">
-      We address the challenges of connecting measurements of a physical system to an underlying theoretical description.
-    </div>
+    <a href="/project/mbdynamics/"  class="summary-link">
+      <div class="article-style">
+        Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -1914,8 +2620,11 @@ <h3 class="article-title mb-0 mt-0">
       <div>
         
 
-  
-  <span>FS in collaboration with Eliska Greplova</span>, <span>Tomoki Hirosawa</span>, <span>Jozef Bucko</span>, <span>František Herman</span>, <span>Rebekka Garreis</span>, <span>Chuyao Tong</span>, <span>Annika Kurzmann</span>, <span>Thomas Ihn</span>, <span>Hideaki Maebashi</span>, <span>Hiroyasu Matsuura</span>, <span>and Masao Ogata</span>
+  <span >
+      FS in Collaboration With Laurent De Forges De Parny</span>, <span >
+      Miguel Bastarrachea</span>, <span >
+      Axel Lode</span>, <span >
+      And Andreas Buchleitner</span>
       </div>
       
     </div>
@@ -1961,10 +2670,8 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.19.024015" target="_blank" rel="noopener">
-    
-    Phys. Rev. Applied
-  </a>
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.mdpi.com/1099-4300/22/4/382" target="_blank" rel="noopener">
+    Entropy</a>
 
   
   
@@ -1977,10 +2684,22 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://journals.jps.jp/doi/abs/10.7566/JPSJ.91.114603" target="_blank" rel="noopener">
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://publikationen.uni-tuebingen.de/xmlui/handle/10900/87664" target="_blank" rel="noopener">
+    bwHPC proceeding</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
     
-    J. Phys. Soc. Jpn.
-  </a>
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://freidok.uni-freiburg.de/data/154730" target="_blank" rel="noopener">
+    Master thesis</a>
 
 
     </div>
@@ -1990,30 +2709,18 @@ <h3 class="article-title mb-0 mt-0">
   <div class="ml-3">
     
     
-    
-    <a href="/project/reconstruction/" >
-      <img src="/project/reconstruction/featured_hub0c7e03472f12c4e5847301dd7a71967_835003_150x0_resize_lanczos_2.png" alt="">
-    </a>
+      
+      
+      <a href="/project/mbdynamics/" >
+        <img src="/project/mbdynamics/featured_hu7f7947d2046583f5302b04408a1eb220_58439_234d3a3c6cd272b12a2fa1b17ae4fbb3.webp" height="150" width="150"
+            alt="Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well " loading="lazy">
+      </a>
     
   </div>
 </div>
 
-</div>
-
-        
-
-      
-
-        
-        
-        
-
-        
-          
-
-
-<div class="col-12 isotope-item js-id-Machine-learning js-id-analytical-predictors js-id-neural-networks js-id-unsupervised-learning">
   
+    
 
 
 
@@ -2021,28 +2728,30 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
-
-  
   
 
 
 
 
 
+
+
   
 
 
-<div class="media stream-item">
+<div class="media stream-item view-compact">
   <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/scattering/" >Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers</a>
+    </div>
 
     
-    <div class="article-style">
-      Data-driven methods based on sample instances of the state of a physical system as a function of the system&rsquo;s parameters.
-    </div>
+    <a href="/project/scattering/"  class="summary-link">
+      <div class="article-style">
+        Numerical optimization of the positions of point scatterers to maximize the total scattering cross section for an incoming plane wave.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -2053,8 +2762,8 @@ <h3 class="article-title mb-0 mt-0">
       <div>
         
 
-  
-  <span>FS in collaboration with Julian Arnold</span>, <span>Flemming Holtorf</span>, <span>Eliska Greplova</span>, <span>Agnes Valenti</span>, <span>Martin Zonda</span>, <span>Axel Lode</span>, <span>Gregor Boschung</span>, <span>Sebastian Huber</span>, <span>and Niels Lörch</span>
+  <span >
+      FS in Collaboration With Felix Eckert and Thomas Wellens</span>
       </div>
       
     </div>
@@ -2071,19 +2780,15 @@ <h3 class="article-title mb-0 mt-0">
 
 
   
-
-
-
-
-
-
-  
-  
     
   
-<a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://github.com/arnoldjulian/Replacing-neural-networks-by-optimal-analytical-predictors-for-the-detection-of-phase-transitions" target="_blank" rel="noopener">
-  Code
-</a>
+
+
+
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/1361-6455/aa92d8/meta" target="_blank" rel="noopener">
+  PDF
+</a>
+
 
 
 
@@ -2097,37 +2802,7 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
-  
-  
-  
-    
-  
-  
-  
-  
-  
-    
-  
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
-    
-    PRE
-  </a>
 
-  
-  
-  
-    
-  
-  
-  
-  
-  
-    
-  
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://iopscience.iop.org/article/10.1088/1367-2630/ab7771/meta" target="_blank" rel="noopener">
-    
-    NJP
-  </a>
 
   
   
@@ -2140,75 +2815,58 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.033052" target="_blank" rel="noopener">
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://freidok.uni-freiburg.de/data/12809" target="_blank" rel="noopener">
+    Bachelor thesis</a>
+
+
+    </div>
     
-    PRResearch
-  </a>
 
-  
-  
-  
+  </div>
+  <div class="ml-3">
     
-  
-  
-  
-  
-  
     
-  
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.12.031044" target="_blank" rel="noopener">
+      
+      
+      <a href="/project/scattering/" >
+        <img src="/project/scattering/featured_huf3ed79a93892e0cfaab1947971615755_22233_71ce5e29cd78133a72f9369788482dfc.webp" height="100" width="150"
+            alt="Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers" loading="lazy">
+      </a>
     
-    PRX
-  </a>
+  </div>
+</div>
 
   
+
   
   
-    
   
+
+</div>
+
+
   
+    </div>
   
+
+  </div>
+</section>
+
   
+
   
-    
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://arxiv.org/abs/2311.09128" target="_blank" rel="noopener">
     
-    NeurIPS Workshop
-  </a>
 
 
-    </div>
-    
 
-  </div>
-  <div class="ml-3">
-    
-    
-    
-    <a href="/project/ml_for_pt/" >
-      <img src="/project/ml_for_pt/featured_hu3194460ba82e7df78553a9c87bbc0c72_237834_150x0_resize_lanczos_2.png" alt="">
-    </a>
-    
-  </div>
-</div>
 
-</div>
 
-        
 
-      
 
-        
-        
-        
 
-        
-          
 
 
-<div class="col-12 isotope-item js-id-Many-body-physics js-id-exact-diagonalization js-id-Bose-Hubbard-model js-id-MCTDH-X">
-  
 
 
 
@@ -2217,46 +2875,20 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
-  
-  
 
 
 
 
 
-  
 
 
-<div class="media stream-item">
-  <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a>
-    </h3>
 
-    
-    <div class="article-style">
-      Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
-    </div>
-    
 
-    <div class="stream-meta article-metadata">
 
-      
 
-      
-      <div>
-        
 
-  
-  <span>FS in collaboration with Laurent de Forges de Parny</span>, <span>Miguel Bastarrachea</span>, <span>Axel Lode</span>, <span>and Andreas Buchleitner</span>
-      </div>
-      
-    </div>
 
-    
-    <div class="btn-links">
-      
 
 
 
@@ -2265,7 +2897,6 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
-  
 
 
 
@@ -2284,88 +2915,85 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
-  
-  
-  
-    
-  
-  
-  
-  
-  
-    
-  
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://www.mdpi.com/1099-4300/22/4/382" target="_blank" rel="noopener">
-    
-    Entropy
-  </a>
+
+
+
+<section id="software" class="home-section wg-collection  "  >
+ <div class="home-section-bg " >
+   
+ </div>
+  <div class="container">
 
   
-  
-  
+    <div class="row  ">
     
-  
-  
-  
-  
-  
+      
+        <div class="section-heading col-12 col-lg-4 mb-3 mb-lg-0 d-flex flex-column align-items-center align-items-lg-start">
+          <h1 class="mb-0">Open source software</h1>
+          
+        </div>
+      
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://publikationen.uni-tuebingen.de/xmlui/handle/10900/87664" target="_blank" rel="noopener">
-    
-    bwHPC proceeding
-  </a>
 
-  
-  
-  
     
+
+
+
+
+
+
+
+
+
+
   
+
+
+
+
+
+
+
+
+
   
   
   
   
-    
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://freidok.uni-freiburg.de/data/154730" target="_blank" rel="noopener">
-    
-    Master thesis
-  </a>
 
 
-    </div>
-    
 
-  </div>
-  <div class="ml-3">
-    
-    
-    
-    <a href="/project/mbdynamics/" >
-      <img src="/project/mbdynamics/featured_hu7f7947d2046583f5302b04408a1eb220_58439_150x0_resize_lanczos_2.png" alt="">
-    </a>
-    
-  </div>
-</div>
 
-</div>
 
-        
 
-      
 
-        
-        
-        
 
-        
-          
 
 
-<div class="col-12 isotope-item js-id-multiple-scattering-theory">
+
+
+
+
+
+
+
+
+
+
+
   
 
 
+
+
+<div class="col-12 col-lg-8">
+
+  
+
+  
+    
 
 
 
@@ -2378,21 +3006,26 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
+
+
+
 
   
 
 
-<div class="media stream-item">
+<div class="media stream-item view-compact">
   <div class="media-body">
 
-    <h3 class="article-title mb-0 mt-0">
-      <a href="/project/scattering/" >Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="https://github.com/gaurav-arya/StochasticAD.jl" target="_blank" rel="noopener">StochasticAD.jl: Automatic Differentiation of Programs with Discrete Randomness</a>
+    </div>
 
     
-    <div class="article-style">
-      Numerical optimization of the positions of point scatterers to maximize the total scattering cross section for an incoming plane wave.
-    </div>
+    <a href="https://github.com/gaurav-arya/StochasticAD.jl" target="_blank" rel="noopener" class="summary-link">
+      <div class="article-style">
+        StochasticAD is an experimental, research package for automatic differentiation (AD) of stochastic programs. It implements AD algorithms for handling programs that can contain discrete randomness. Joint work with: <a href="https://github.com/gaurav-arya" target="_blank" rel="noopener">Gaurav Arya</a>, <a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, and <a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -2400,12 +3033,38 @@ <h3 class="article-title mb-0 mt-0">
       
 
       
-      <div>
         
 
+
+<div class="article-metadata">
+
   
-  <span>FS in collaboration with Felix Eckert and Thomas Wellens</span>
-      </div>
+  
+
+  
+  <span class="article-date">
+    
+    
+      
+    
+    Jan 16, 2023
+  </span>
+  
+
+  
+
+  
+
+  
+  
+  
+  
+
+  
+  
+
+</div>
+
       
     </div>
 
@@ -2421,14 +3080,8 @@ <h3 class="article-title mb-0 mt-0">
 
 
   
-    
-  
-
 
 
-<a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://iopscience.iop.org/article/10.1088/1361-6455/aa92d8/meta" target="_blank" rel="noopener">
-  PDF
-</a>
 
 
 
@@ -2456,10 +3109,8 @@ <h3 class="article-title mb-0 mt-0">
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://freidok.uni-freiburg.de/data/12809" target="_blank" rel="noopener">
-    
-    Bachelor thesis
-  </a>
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2210.08572" target="_blank" rel="noopener">
+    NeurIPS 2022 paper</a>
 
 
     </div>
@@ -2469,49 +3120,72 @@ <h3 class="article-title mb-0 mt-0">
   <div class="ml-3">
     
     
-    
-    <a href="/project/scattering/" >
-      <img src="/project/scattering/featured_huf3ed79a93892e0cfaab1947971615755_22233_150x0_resize_lanczos_2.png" alt="">
-    </a>
-    
   </div>
 </div>
 
-</div>
+  
+    
+
 
-        
 
-      
-    </div>
 
-  </div>
-</div>
 
-    </div>
-  </section>
 
   
   
+
+
+
+
+
+
+
+
   
 
-  
 
-  
+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl" target="_blank" rel="noopener">AbstractDifferentiation.jl: Backend-Agnostic Differentiable Programming in Julia</a>
+    </div>
+
+    
+    <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl" target="_blank" rel="noopener" class="summary-link">
+      <div class="article-style">
+        Automatized generation of an extensive, unified, user-facing API for any AD package. Enables easy switching and composing between AD implementations. Joint work with <a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a> and <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/graphs/contributors" target="_blank" rel="noopener">other contributors</a>.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
 
-  
+      
+        
 
-  
 
-  
+<div class="article-metadata">
 
   
   
 
   
+  <span class="article-date">
+    
+    
+      
+    
+    Dec 3, 2021
+  </span>
   
+
   
 
   
+
   
   
   
@@ -2520,10 +3194,14 @@ <h3 class="article-title mb-0 mt-0">
   
   
 
-  <section id="software" class="home-section wg-portfolio   "  >
-    <div class="container">
+</div>
+
       
+    </div>
 
+    
+    <div class="btn-links">
+      
 
 
 
@@ -2532,39 +3210,52 @@ <h3 class="article-title mb-0 mt-0">
 
 
 
+  
+
+
+
+
+
 
-<div class="row">
-  <div class="col-12 col-lg-4 section-heading">
 
-    <h1>Open source software</h1>
-    
 
-  </div>
-  <div class="col-12 col-lg-8">
 
 
 
-    
 
-    
 
-    <div class=" row js-layout-row ">
-      
 
-        
-        
-        
-          
-          
-        
 
-        
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-<div class="col-12 isotope-item ">
+
+  
+  
+  
+    
+  
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+  
   
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2109.12449" target="_blank" rel="noopener">
+    NeurIPS 2021 workshop</a>
+
+
+    </div>
+    
+
+  </div>
+  <div class="ml-3">
+    
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-<div class="media stream-item">
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-    <h3 class="article-title mb-0 mt-0">
-      <a href="https://github.com/gaurav-arya/StochasticAD.jl" target="_blank" rel="noopener">StochasticAD.jl: Automatic Differentiation of Programs with Discrete Randomness</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="https://sciml.ai/" target="_blank" rel="noopener">SciML Scientific Machine Learning Software</a>
+    </div>
 
     
-    <div class="article-style">
-      StochasticAD is an experimental, research package for automatic differentiation (AD) of stochastic programs. It implements AD algorithms for handling programs that can contain discrete randomness. Joint work with: <a href="https://github.com/gaurav-arya">Gaurav Arya</a>, <a href="https://github.com/mschauer">Moritz Schauer</a>, and <a href="https://github.com/ChrisRackauckas">Chris Rackauckas</a>.
-    </div>
+    <a href="https://sciml.ai/" target="_blank" rel="noopener" class="summary-link">
+      <div class="article-style">
+        Contributions to the SciML ecosystem in Julia, especially the DiffEqSensitivity.jl package for sensitivity analysis utilities, the StochasticDiffEq.jl package for stochastic differential equations solvers, and the DiffEqNoiseProcess package for tools to develop noise processes for differential equations.
+      </div>
+    </a>
     
 
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-    Jan 16, 2023
+    Mar 10, 2021
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+
+
   
 
 
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+<div class="media stream-item view-compact">
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-    <h3 class="article-title mb-0 mt-0">
-      <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl" target="_blank" rel="noopener">AbstractDifferentiation.jl: Backend-Agnostic Differentiable Programming in Julia</a>
-    </h3>
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="https://github.com/mschauer/MitosisStochasticDiffEq.jl" target="_blank" rel="noopener">MitosisStochasticDiffEq.jl</a>
+    </div>
 
     
-    <div class="article-style">
-      Automatized generation of an extensive, unified, user-facing API for any AD package. Enables easy switching and composing between AD implementations. Joint work with <a href="https://github.com/mohamed82008">Mohamed Tarek</a> and <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/graphs/contributors">other contributors</a>.
-    </div>
+    <a href="https://github.com/mschauer/MitosisStochasticDiffEq.jl" target="_blank" rel="noopener" class="summary-link">
+      <div class="article-style">
+        Implementation of the backward filter and the forward change of measure of the Automatic Backward Filtering Forward Guiding paradigm. Joint work with <a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>.
+      </div>
+    </a>
     
 
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-      <a href="https://sciml.ai/" target="_blank" rel="noopener">SciML Scientific Machine Learning Software</a>
-    </h3>
 
-    
-    <div class="article-style">
-      Contributions to the SciML ecosystem in Julia, especially the DiffEqSensitivity.jl package for sensitivity analysis utilities, the StochasticDiffEq.jl package for stochastic differential equations solvers, and the DiffEqNoiseProcess package for tools to develop noise processes for differential equations.
-    </div>
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+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
 
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diff --git a/index.json b/index.json
index 9f39abb..1db994f 100644
--- a/index.json
+++ b/index.json
@@ -1 +1,30 @@
-[{"authors":["admin"],"categories":null,"content":"I am a postdoc in the Julia Lab located in the Computer Science and Artificial Intelligence Laboratory (CSAIL) at the Massachusetts Institute of Technology (MIT). My research is focused on Scientific Machine Learning (SciML). I completed my PhD in physics in the Bruder group within the \u0026ldquo;Quantum Computing and Quantum Technology\u0026rdquo; PhD school at the University of Basel. During my PhD, I participated in the Google Summer of Code (GSoC) 2020 and 2021 programs with the projects \u0026ldquo; High weak order stochastic differential equation solvers and their utility in neural stochastic differential equations\u0026rdquo; within the Julia Language organization and \u0026ldquo; Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis\u0026rdquo; within the NumFocus organization, supervised by Chris Rackauckas, Moritz Schauer, Mohamed Tarek, and Yingbo Ma. Since 2020, I am a member of the SciML open source software organization for scientific machine learning.\n","date":-62135596800,"expirydate":-62135596800,"kind":"term","lang":"en","lastmod":-62135596800,"objectID":"2525497d367e79493fd32b198b28f040","permalink":"https://frankschae.github.io/author/frank-schafer/","publishdate":"0001-01-01T00:00:00Z","relpermalink":"/author/frank-schafer/","section":"authors","summary":"I am a postdoc in the Julia Lab located in the Computer Science and Artificial Intelligence Laboratory (CSAIL) at the Massachusetts Institute of Technology (MIT). My research is focused on Scientific Machine Learning (SciML).","tags":null,"title":"Frank Schäfer","type":"authors"},{"authors":["FS in collaboration with Gaurav Arya","Moritz Schauer","Ruben Seyer","Alex Lew","Mathieu Huot","Kartik Chandra","Vikash Mansinghka","Jonathan Ragan-Kelley","Chris Rackauckas"],"categories":[],"content":"Automatic differentiation (AD) has become ubiquitous throughout scientific computing and deep learning. However, AD systems have been restricted to the subset of programs that have a continuous dependence on parameters. Programs that have discrete stochastic behaviors governed by distribution parameters, such as flipping a coin with probability p of being heads, pose a challenge to these systems. In this work we develop a new AD methodology for programs with discrete randomness. We demonstrate how this method gives an unbiased and low-variance estimator.\n","date":1705333764,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1705333764,"objectID":"fd4013b59c325e7176a3ea5f9c2d78c3","permalink":"https://frankschae.github.io/project/stochasticad/","publishdate":"2024-01-15T10:49:24-05:00","relpermalink":"/project/stochasticad/","section":"project","summary":"We develop and implement AD algorithms for handling programs that can contain discrete randomness.","tags":[],"title":"Automatic Differentiation of Programs with Discrete Randomness","type":"project"},{"authors":["FS in collaboration with Flemming Holtorf","Julian Arnold","Chris Rackauckas","and Alan Edelman"],"categories":[],"content":"Control of devices at the quantum level holds enormous potential for current and future applications in the field of quantum information science. However, due to the nonlinear and stochastic nature of quantum systems under continuous observation, analytical solutions to all but the simplest quantum control problems remain unknown. In this project, we present a convex optimization framework to compute informative bounds on the best attainable control performance. Since our approach provides an under-approximator for the value function, we can use it directly to construct near-optimal heuristic controllers as demonstrated for a qubit subjected to homodyne detection and photon counting.\n","date":1682803609,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1682803609,"objectID":"d1e077164ce2a89141c07c546777d1ea","permalink":"https://frankschae.github.io/project/controlbounds/","publishdate":"2023-04-29T17:26:49-04:00","relpermalink":"/project/controlbounds/","section":"project","summary":"We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.","tags":["quantum control","convex optimization","stochastic optimal control","sum-of-squares"],"title":"Performance Bounds for Quantum Control","type":"project"},{"authors":[],"categories":[],"content":"","date":1673824837,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1673824837,"objectID":"91750f0e293b4386e71240dea8d36441","permalink":"https://frankschae.github.io/software/stochasticad/","publishdate":"2023-01-16T00:20:37+01:00","relpermalink":"/software/stochasticad/","section":"software","summary":"StochasticAD is an experimental, research package for automatic differentiation (AD) of stochastic programs. It implements AD algorithms for handling programs that can contain discrete randomness. Joint work with: [Gaurav Arya](https://github.com/gaurav-arya), [Moritz Schauer](https://github.com/mschauer), and [Chris Rackauckas](https://github.com/ChrisRackauckas).","tags":[],"title":"StochasticAD.jl: Automatic Differentiation of Programs with Discrete Randomness","type":"software"},{"authors":[],"categories":[],"content":"","date":1638487237,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1638487237,"objectID":"dd74b30c2376d17e758c3568ec2a62ad","permalink":"https://frankschae.github.io/software/abstractdifferentiation/","publishdate":"2021-12-03T00:20:37+01:00","relpermalink":"/software/abstractdifferentiation/","section":"software","summary":"Automatized generation of an extensive, unified, user-facing API for any AD package. Enables easy switching and composing between AD implementations. Joint work with [Mohamed Tarek](https://github.com/mohamed82008) and [other contributors](https://github.com/JuliaDiff/AbstractDifferentiation.jl/graphs/contributors).","tags":[],"title":"AbstractDifferentiation.jl: Backend-Agnostic Differentiable Programming in Julia","type":"software"},{"authors":[],"categories":[],"content":"Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.\nPossible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available on the GSoC page.\nBlog posts The following blog posts describe the work throughout the GSoC period in more detail:\n  Neural Hybrid Differential Equations  Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems  Sensitivity Analysis of Hybrid Differential Equations  AbstractDifferentiation.jl for AD-backend agnostic code  Docs Documentation with respect to the adjoint sensitivity tools will be available on the local sensitivity analysis and on the control of automatic differentiation choices pages.\nAchievements Below is a list of PRs in the various repositories in chronological order.\nDiffEqSensitivity.jl Merged:\n  Add additive noise downstream test for DiffEqFlux  DiscreteCallback fixes  Allow for changes of p in callbacks  Fix for using the correct uleft/pleft in continuous callback  Fix broadcasting error on steady state adjoint  Forward Least Squares Shadowing (LSS)  Adjoint-mode for the LSS method  concrete_solve dispatch for LSS methods  Non-Intrusive Least Square Shadowing (NILSS)  concrete_solve for NILSS  Remove allocation in NILSS  Handle additional callback case  State-dependent Continuous Callbacks for BacksolveAdjoint  QuadratureAdjoint() for ContinuousCallback  More tests for Neural ODEs with callbacks for different sensitivity algorithms  Support for PeriodicCallbacks in continuous adjoint methods  AbstractDifferentiation.jl Merged:\n  Fixes gradient, Jacobian, Hessian, and vjp tests  Open:\n  Add ForwardDiff and Zygote  OrdinaryDiffEq.jl Merged:\n  Fix discrete reverse mode for some standard controllers  DiffEqCallbacks.jl Merged:\n  Introduce a PeriodicCallbackAffect struct  SteadyStateDiffEq.jl Merged:\n  Convert alg.tspan to type of prob.u0  ChainRules.jl Merged:\n  Do not differentiate through the construction of BitArray  Use splatting in BitArray  DiffEqNoiseProcess.jl Merged:\n  Allow solvers to use Noise Grid with SVectors  StochasticDiffEq.jl Merged:\n  Remove Ihat2 matrix from weak solvers  DiffEqDocs.jl Merged:\n  Small typo on plot page  Add docs for shadowing methods  Future work Besides the implementation of more shadowing methods, such as\n  NILSAS,  FD-NILSS, or  Fast linear response,  we are planning to\n benchmark the new adjoints, refine the AbstractDifferentiation.jl package and use it within DiffEqSensitivity.jl, add more docs and examples.  If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the Julia language discourse.\nAcknowledgement Many thanks to my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times. I would also like to thank Christoph Bruder, Julian Arnold, and Martin Koppenhöfer for helpful comments on my blog posts. Special thanks to Michael Poli and Stefano Massaroli for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google\u0026rsquo;s open source program for funding this experience!\n","date":1628883705,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1628883705,"objectID":"295f158a34d7a2a8bb042950210377c3","permalink":"https://frankschae.github.io/post/gsoc-2021/","publishdate":"2021-08-13T21:41:45+02:00","relpermalink":"/post/gsoc-2021/","section":"post","summary":"Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.","tags":["GSoC 2021"],"title":"Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis","type":"post"},{"authors":[],"categories":[],"content":" Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris' article3 for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance and accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified4, for example, by focusing on learning only the parts of the model that are actually missing4 5. In the context of quantum control, we have demonstrated the power of this framework for closed6 and open quantum systems7.\n∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, a large variety of software implementations with different pros and cons exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by ChainRules.jl empowers this idea of a Glue AD where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.\n Mohamed Tarek has started to implement a high level API for differentiation that unifies the APIs of all the AD packages in the Julia ecosystem. Ultimately, the API of our new package, AbstractDifferentiation.jl, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that DiffEqSensitivity.jl within the SciML software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I\u0026rsquo;ve started to fix remaining errors of the initial PR.\nThe interested reader is encouraged to look at Mohamed\u0026rsquo;s first PR for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.\nOptimization of the Rosenbrock function The Rosenbrock function is defined by\n$$ g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2. $$\nThe function $g$ has a global minimum at $(x_1^\\star, x_2^\\star)= (a, a^2)$ with $g(x_1^\\star, x_2^\\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.\nLet us now implement the Gauss–Newton algorithm to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\\bf{x}}=(x_1,\\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\\dots, f_M)$\n$$ S({\\bf x}) = \\frac{1}{2} \\sum_{i=1}^M f_i({\\bf x})^2. $$\nStarting from an initial guess ${\\bf x_0}$ for the minimum, the method runs through the iterations\n$$ {\\bf x}^{k+1} = {\\bf x}^k - \\alpha_k \\left(J^T J \\right)^{-1} J^T f({\\bf x}^k), $$ where $J$ is the Jacobian matrix at ${\\bf{x}}^k$ and $\\alpha_k$ is the step length determined via a line search subroutine.\nThe following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\\bf x^\\star}=(1,1)$ and our initial guess ${\\bf x_0}=(0,-0.1)$.\nusing Pkg path = @__DIR__ cd(path); Pkg.activate(\u0026quot;.\u0026quot;); Pkg.instantiate() ## AbstractDifferentiation is not released yet!! using AbstractDifferentiation using Test, LinearAlgebra using FiniteDifferences, ForwardDiff, Zygote using Enzyme, UnPack using Plots, LaTeXStrings # using Diffractor: ∂⃖¹ ## Diffractor needs \u0026gt;julia@1.6 ## Rosenbrock function # R: R^2 -\u0026gt; R: x -\u0026gt; (a-x₁)² + b(x₂-x₁²)² g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2 # visualization p = [1.0,100.0] x₀ = [0.0,-0.1] xopt = [1.0,1.0] do_plot = true if do_plot x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5 z = Surface((x₁,x₂)-\u0026gt;g([x₁,x₂],p), x₁, x₂) pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true, labelfontsize=20,camera = (3,50), xlabel = L\u0026quot;x_1\u0026quot;, ylabel = L\u0026quot;x_2\u0026quot;) pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp), labelfontsize=20, xlabel = L\u0026quot;x_1\u0026quot;, ylabel = L\u0026quot;x_2\u0026quot;) scatter!(pl2, [(x₀[1],x₀[2])], label=L\u0026quot;x_0\u0026quot;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green) scatter!(pl2, [(xopt[1],xopt[2])],label=L\u0026quot;x^\\star\u0026quot;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red) pl = plot(pl1,pl2, layout=(2,1)) savefig(pl, \u0026quot;Rosenbrock.png\u0026quot;) end    To apply the Gauss-Newton algorithm to the Rosenbrock function $g$, we first cast $g$ into an appropriate form fulfilling $S({\\bf x})$, i.e., we use:\n$$ f:\\mathbb{R}^2\\rightarrow\\mathbb{R}^2: {\\bf x} \\mapsto \\begin{pmatrix} f_1({\\bf x}) \\\\\nf_2({\\bf x}) \\\\\n\\end{pmatrix} = \\begin{pmatrix} \\sqrt{2}(a-x_1) \\\\\n\\sqrt{2b}(x_2-x_1^2)\\\\\n\\end{pmatrix}, $$\ninstead of $g$. We can easily compute the Jacobian of $f$ manually\n$$ J = \\begin{pmatrix} -\\sqrt{2} \u0026amp; 0 \\\\\n-2x_1\\sqrt{2b} \u0026amp; \\sqrt{2b} \\\\\n\\end{pmatrix}. $$\nWe can then implement a (simple, non-optimized) version of the Gauss-Newton algorithm as follows.\n# bring Rosenbrock function into the form \u0026quot;sum of squares of functions\u0026quot; f1(x,p) = convert(eltype(x),sqrt(2))*(p[1]-x[1]) f2(x,p) = convert(eltype(x),sqrt(2*p[2]))*(x[2]-x[1]^2) f(x,p) = [f1(x,p),f2(x,p)] function f(res,x,p) # Enzyme works with inplace functions res[1] = f1(x,p) res[2] = f2(x,p) return nothing end ## manually pre-defined Jacobian function Jacobian(x,p) [-convert(eltype(x),sqrt(2)) 0 -2*x[1]*convert(eltype(x),sqrt(2*p[2])) convert(eltype(x),sqrt(2*p[2]))] end ## Gauss-Newton scheme function GaussNewton!(xs, x, p; maxiter=8, backend=nothing) for i=1:maxiter x = step(x, p, backend) @info i @show x push!(xs, x) end return xs, x end done(x,x2,p) = g(x2,p) \u0026lt; g(x,p) function step(x, p, backend::Nothing, α=1//1) x2 = deepcopy(x) while !done(x,x2,p) J = Jacobian(x,p) d = -inv(J'*J)*J'*f(x,p) copyto!(x2,x + α*d) α = α//2 end return x2 end  When we run the algorithm, we find the global minimum after about the 7th iteration.\nxs = [x₀] GaussNewton!(xs, x₀, p)  # output: [ Info: 1 ] x = [0.125, -0.08750000000000001] [ Info: 2 ] x = [0.234375, -0.047265625000000006] [ Info: 3 ] x = [0.4257812499999995, 0.06800537109374968] [ Info: 4 ] x = [0.5693359374999986, 0.21857223510742047] [ Info: 5 ] x = [0.784667968749996, 0.5165503501892037] [ Info: 6 ] x = [0.9999999999999961, 0.9536321163177449] [ Info: 7 ] x = [0.9999999999999989, 0.9999999999999999] [ Info: 8 ] x = [1.0, 1.0]  If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of FiniteDifferences.jl as a new primitive operation.\n## FiniteDifferences struct FDMBackend{A} \u0026lt;: AD.AbstractFiniteDifference alg::A end FDMBackend() = FDMBackend(central_fdm(5, 1)) const fdm_backend = FDMBackend() # Minimal interface AD.@primitive function jacobian(ab::FDMBackend, f, xs...) return FiniteDifferences.jacobian(ab.alg, f, xs...) end # AD Jacobian returns tuple # df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1] # df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2] @test AD.jacobian(fdm_backend, x-\u0026gt;f(x,p), x₀)[1] ≈ Jacobian(x₀, p) @test AD.jacobian(fdm_backend, f, x₀, p)[1] ≈ Jacobian(x₀, p)  After overloading the step function, we can run the Gauss-Newton algorithm as follows:\nfunction step(x, p, backend, α=1//1) x2 = deepcopy(x) while !done(x,x2,p) J = AD.jacobian(backend, f, x, p)[1] d = -inv(J'*J)*J'*f(x,p) copyto!(x2,x + α*d) α = α//2 end return x2 end xs = [x₀] GaussNewton!(xs, x₀, p, backend=fdm_backend)  If we want to use reverse-mode AD instead, for example via Zygote.jl, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.\n## Zygote struct ZygoteBackend \u0026lt;: AD.AbstractReverseMode end const zygote_backend = ZygoteBackend() AD.@primitive function pullback_function(ab::ZygoteBackend, f, xs...) return function (vs) # Supports only single output _, back = Zygote.pullback(f, xs...) if vs isa AbstractVector back(vs) else @assert length(vs) == 1 back(vs[1]) end end end ## @test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(zygote_backend, f, x₀, p)) xs = [x₀] GaussNewton!(xs, x₀, p, backend=zygote_backend)  Typically, reverse-mode AD is only beneficial for functions $f:\\mathbb{R}^N\\rightarrow\\mathbb{R}^M$ where $M \\ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD ( ForwardDiff.jl)\n## ForwardDiff struct ForwardDiffBackend \u0026lt;: AD.AbstractForwardMode end const forwarddiff_backend = ForwardDiffBackend() AD.@primitive function pushforward_function(ab::ForwardDiffBackend, f, xs...) # jvp = f'(x)*v, i.e., differentiate f(x + h*v) wrt h at 0 return function (vs) if xs isa Tuple @assert length(xs) \u0026lt;= 2 if length(xs) == 1 (ForwardDiff.derivative(h-\u0026gt;f(xs[1]+h*vs[1]),0),) else ForwardDiff.derivative(h-\u0026gt;f(xs[1]+h*vs[1], xs[2]+h*vs[2]),0) end else ForwardDiff.derivative(h-\u0026gt;f(xs+h*vs),0) end end end ## @test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(forwarddiff_backend, f, x₀, p)) xs = [x₀] GaussNewton!(xs, x₀, p, backend=forwarddiff_backend)  where we have used that the Jacobian-vector product $f'(x)v$, i.e., the primitives of forward-mode AD, can be computed by differentiating $f(x + hv)$ with respect to $h$ at 0.\nMany AD packages, such as Zygote, have troubles with mutating functions. Enzyme.jl is one of the exceptions. Additionally, it is very fast and has further improved the performance of the adjoints implemented within the DiffEqSensitivity package.\n## Enzyme struct EnzymeBackend{T1,T2,T3,T4} \u0026lt;: AD.AbstractReverseMode out::T1 λ::T2 ∂f_∂x::T3 ∂f_∂p::T4 end out = zero(x₀) λ = zero(x₀) ∂f_∂x = zero(x₀) ∂f_∂p = zero(p) const enzyme_backend = EnzymeBackend(out,λ,∂f_∂x,∂f_∂p) AD.@primitive function pullback_function(ab::EnzymeBackend, f, xs...) return function (vs) # enzyme works only with inplace functions if !(vs isa AbstractVector) @assert length(vs) == 1 # Supports only single output vs = vs[1] end if xs isa Tuple @assert length(xs) == 2 # hard-coded for use case with two inputs x₀ = xs[1] p = xs[2] end @unpack out, λ, ∂f_∂x, ∂f_∂p = ab # cached in the struct, could also be created in here ∂f_∂x .*= false ∂f_∂p .*= false out .*= false copyto!(λ, vs) autodiff(Duplicated(out, λ), Duplicated(x₀, ∂f_∂x), Duplicated(p, ∂f_∂p)) do _out,_x, _p f(_out,_x,_p) end return (∂f_∂x,∂f_∂p) end end AD.isinplace(ab::EnzymeBackend) = true AD.primalvalue(ab::EnzymeBackend, nothing, f, xs) = (f(ab.out,xs...);return ab.out) ## @test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(enzyme_backend, f, x₀, p)) xs = [x₀] GaussNewton!(xs, x₀, p, backend=enzyme_backend)  Note that we have declared the Enzyme backend as inplace (which is important for internal control flow) and specified a primalvalue function returning the primal value of the forward pass.\nSome current glitches First, the push forward of a tuple of vectors, e.g., $(v_1, v_2)$, for a function with several input arguments is currently ambiguous. While AD.jacobian primitives and AD.pullback_function primitives interpret the push forward of our $f$ function as\n$$ \\left(\\frac{\\partial f(x_0,p)}{\\partial x} v_1 , \\frac{\\partial f(x_0,p)}{\\partial p} v_2 \\right), $$\nAD.pushforward_function primitives compute\n$$ \\frac{\\partial f(x_0,p)}{\\partial x} v_1 + \\frac{\\partial f(x_0,p)}{\\partial p} v_2. $$\n# pushforward_function wrt to multiple vectors is currently ambiguous vs = (randn(2), randn(2)) res1 = AD.pushforward_function(fdm_backend, f, x₀, p)(vs) res2 = AD.pushforward_function(forwarddiff_backend, f, x₀, p)(vs) @test res2 ≈ res1[1] + res1[2]  Thus, we currently solve this issue by augmenting the input in the case of AD.pushforward_function primitives.\nres2a = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((vs[1], zero(vs[2]))) res2b = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((zero(vs[1]), vs[2])) @test res2a ≈ res1[1] @test res2b ≈ res1[2]  The plural \u0026ldquo;primitives\u0026rdquo; is used here because we may have different pushforward_function primitives for different backends. For instance, we can define an additional pushforward_function primitive for FiniteDifferences by:\nstruct FDMBackend2{A} \u0026lt;: AD.AbstractFiniteDifference alg::A end FDMBackend2() = FDMBackend2(central_fdm(5, 1)) const fdm_backend2 = FDMBackend2() AD.@primitive function pushforward_function(ab::FDMBackend2, f, xs...) return function (vs) FDM.jvp(ab.alg, f, tuple.(xs, vs)...) end end  Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the Hessian function.\n# Hessian only defined with respect to single input variable @test_throws AssertionError H1 = AD.hessian(forwarddiff_backend, g, x₀, p) H1 = AD.hessian(forwarddiff_backend, x-\u0026gt;g(x,p), x₀) H2 = AD.hessian(forwarddiff_backend, p-\u0026gt;g(x₀,p), p)  Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a HigherOrderBackend that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).\n# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote) backends = AD.HigherOrderBackend((forwarddiff_backend,zygote_backend)) H3 = AD.hessian(backends, x-\u0026gt;g(x,p), x₀)  Outlook There are many other use cases, e.g.,\n  Sensitivity analysis of differential equations requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods. The Newton–Raphson method for rootfinding requires the gradient in the case of scalar function $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\\mathbb{R}^N\\rightarrow\\mathbb{R}^N$. The Newton method in optimization requires the computation of the Hessian.  AbstractDifferentiation.jl is by no means complete yet. We are still in the very early stages, but we hope to make significant progress in the coming weeks. Some of the next steps are:\n fixing remaining bugs, e.g., with respect to the computation of the Hessian and adding AD/Finite Differentiation packages such as Diffractor.  If you have any questions or comments, please don’t hesitate to contact me!\n  Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019). \u0026#x21a9;\u0026#xfe0e;\n Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012). \u0026#x21a9;\u0026#xfe0e;\n Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019). \u0026#x21a9;\u0026#xfe0e;\n Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). \u0026#x21a9;\u0026#xfe0e;\n Raj Dandekar, Chris Rackauckas, et al., Patterns 1, 100145 (2020). \u0026#x21a9;\u0026#xfe0e;\n Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. 1, 035009 (2020). \u0026#x21a9;\u0026#xfe0e;\n Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. 2, 035004 (2021). \u0026#x21a9;\u0026#xfe0e;\n   ","date":1627812197,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1627812197,"objectID":"d9ff4a599f5556fadd3c7cc978b481a3","permalink":"https://frankschae.github.io/post/abstract_differentiation/","publishdate":"2021-08-01T12:03:17+02:00","relpermalink":"/post/abstract_differentiation/","section":"post","summary":"Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris' article3 for a detailed examination.","tags":["GSoC 2021","julia","Automatic Differentiation","AbstractDifferentiation.jl"],"title":"AbstractDifferentiation.jl for AD-backend agnostic code ","type":"post"},{"authors":["Frank Schäfer and Moritz Schauer"],"categories":[],"content":"In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations1. We highlight differences between explicit2 and implicit events3 4. As a paradigmatic example, we consider a bouncing ball described by the ODE\n$$ \\begin{aligned} \\text{d}z(t) \u0026amp;= v(t) \\text{d}t, \\\\\n\\text{d}v(t) \u0026amp;= -\\mathrm g\\thinspace \\text{d}t \\end{aligned}\n$$\nwith initial condition\n$$ \\begin{aligned} z(t=0) \u0026amp;= z_0 = 5, \\\\\nv(t=0) \u0026amp;= v_0 = -0.1. \\end{aligned}\n$$\nThe initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.\nSecond, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\\gamma=0.8$ ( coefficient of restitution) that accounts for a imperfect elastic bounce on the ground.\nWhen ignoring the bounces, we can straightforwardly integrate the ODE analytically\n$$ \\begin{aligned} z(t) \u0026amp;= z_0 + v_0 t - \\frac{\\mathrm g}{2} t^2, \\\\\nv(t) \u0026amp;= v_0 - \\mathrm g\\thinspace t \\end{aligned}\n$$\nor numerically using the OrdinaryDiffEq package from the SciML ecosystem.\n### simulate forward using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity using Plots, LaTeXStrings # dynamics function f(du,u,p,t) du[1] = u[2] du[2] = -p[1] end # parameters and solve z0 = 5.0 v0 = -0.1 t0 = 0.0 tend = 1.9 g = 10 γ = 0.8 u0 = [z0,v0] tspan = (t0,tend) p = [g, γ] prob = ODEProblem(f,u0,tspan,p) # plot forward trajectory sol = solve(prob,Tsit5(),saveat=0.1) pl = plot(sol, label = [\u0026quot;z(t)\u0026quot; \u0026quot;v(t)\u0026quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = \u0026quot;t\u0026quot;, legend=:bottomleft) hline!(pl, [0.0], label=false, color=\u0026quot;black\u0026quot;) savefig(pl,\u0026quot;BB_forward_no_bounce.png\u0026quot;)    Of course, this way the ball continues to fall through the barrier at $z=0$.\nForward simulation with events At time $\\tau$ around $\\tau \\approx 1$, the ball hits the ground $z(\\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\\tau$ with new initial conditions\n$$ \\begin{aligned}\nz({\\tau}) \u0026amp;= z(\\tau-) ,\\\\\nv({\\tau})\u0026amp;= -\\gamma v(\\tau-) , \\end{aligned}\n$$\nso that there is a jump in the velocity at the event time: the velocity right before the bounce, the left limit $v(\\tau-)$, and the velocity with which the ball continues its movement after the bounce $v(\\tau)$, are different.\nGiven our analytical solution for the state as a function of time, we can easily compute the event time $\\tau$ in terms of the initial condition and parameters as\n$$ \\tau = \\frac{v_0 + \\sqrt{v_0^2 + 2 \\mathrm g z_0}}{\\mathrm g}. $$\nExplicit events We can define the bounce of the ball as an explicit event by inserting the values of the initial condition and the parameters into the formula for $\\tau$. We obtain\n$$ \\tau = 0.99005. $$\nThe full explicit trajectory $z_{\\rm exp}(t) = z(t)$ is determined by\n$$ z(t) = \\begin{cases} z_0 + v_0 t - \\dfrac{\\mathrm g}{2} t^2 ,\u0026amp; \\forall t \u0026lt; \\tau, \\\\\n-0.4901 \\mathrm g - 0.5 \\mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\\\\n\\quad - (-0.99005 + t) (-0.99005 \\mathrm g + v_0)\\gamma ,\u0026amp; \\forall t \\ge \\tau, \\end{cases} $$\nwhere we used\n$$ \\begin{aligned}\nz({\\tau})\u0026amp;= z_0 + 0.99005 v_0 -0.4901 \\mathrm g, \\\\\nv({\\tau})\u0026amp;= -\\gamma v({\\tau-}) = -\\gamma(v_0 - 0.99005 \\mathrm g) . \\end{aligned}\n$$\nHere the change in state $(z,v)$ at the event time is defined with the help of an affect function\n$$ a(z,v) = (z, -\\gamma v). $$\nNumerically, we use a DiscreteCallback in this case to simulate the system.\n# solve with DiscreteCallback (explicit event) tstar = (v0 + sqrt(v0^2+2*z0*g))/g condition1(u,t,integrator) = (t == tstar) affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2] cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true)) sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar])  Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.\nImplicit events The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)\n$$ g(z,v,p,t), $$\nwhere an event occurs at time $\\tau$ if $g(z(\\tau),v(\\tau),p,\\tau) = 0$. We have already used this condition to define our impact time $\\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time $g(z,v,p,t) = 0$.\nAs in the previous case, we can analytically compute the full trajectory of the ball. By substituting the formula for $\\tau$ we have at the event time\n\\begin{aligned} z({\\tau})\u0026amp;= 0, \\\\\nv({\\tau}-)\u0026amp;= - \\sqrt{v_0^2 + 2 \\mathrm g z_0} \\end{aligned}\nfor the left limit and\n\\begin{aligned} v({\\tau})\u0026amp;= \\gamma \\sqrt{v_0^2 + 2 \\mathrm g z_0} \\end{aligned}\nright after the bounce. Thus, the full trajectory $z_{\\rm imp}(t) = z(t)$ is given by\n$$ (\\star) \\quad z(t) = \\begin{cases} z_0 + v_0 t - \\dfrac{\\mathrm g}{2} t^2 ,\u0026amp; \\forall t \u0026lt; \\tau ,\\\\\n-\\dfrac{-\\mathrm g t + v_0 + \\sqrt{v_0^2 + 2 \\mathrm g z_0}}{2 \\mathrm g} \\\\\n\\quad\\cdot \\space (-\\mathrm g t + v_0 + \\sqrt{v_0^2 + 2 \\mathrm g z_0} (1 + 2 \\gamma)), \u0026amp; \\forall t \\ge \\tau. \\end{cases} $$\nThis is correct even if one substitutes, e.g., a value with higher precision $\\mathrm g = 9.81$ for the gravitation constant.\nNumerically, we use a ContinuousCallback in this case.\n# solve with ContinuousCallback (implicit event) condition2(u,t,integrator) = u[1] # Event happens when condition2(u,t,integrator) == 0 cb2 = ContinuousCallback(condition2,affect!,save_positions=(true,true)) sol2 = solve(prob,Tsit5(),callback=cb2,saveat=0.1)  We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).\n# plot forward trajectory pl1 = plot(sol1, label = [\u0026quot;z(t)\u0026quot; \u0026quot;v(t)\u0026quot;], title=\u0026quot;explicit event\u0026quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = \u0026quot;t\u0026quot;, legend=:bottomright) pl2 = plot(sol2, label = [\u0026quot;z(t)\u0026quot; \u0026quot;v(t)\u0026quot;], title=\u0026quot;implicit event\u0026quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = \u0026quot;t\u0026quot;, legend=:bottomright) hline!(pl1, [0.0], label=false, color=\u0026quot;black\u0026quot;) hline!(pl2, [0.0], label=false, color=\u0026quot;black\u0026quot;) pl = plot(pl1,pl2) savefig(pl,\u0026quot;BB_forward.png\u0026quot;)    In addition, the implicitly defined impact time via the ContinuousCallback also changes appropriately when changing the initial conditions or the parameters, for example when using $\\mathrm g = 9.81$ for the gravitation constant. In other words, the event time $\\tau=\\tau(p,z_0,v_0,t_0)$ is a function of the parameters and initial conditions, and is implicitly defined by the event condition.\nSuppose we let the ball drop from a somewhat higher position now. Does an increase in height $z$ at $t=0$ give an increase or decrease in height at the end time $t_\\text{end}=1.9$? This is something we can answer with sensitivity analysis. For example if we increase the height by (a fraction of) one unit then using $(\\star)$\n$$ \\frac{\\text{d} z(t_\\text{end})}{\\text{d} z_0} = 0.84, $$\nmeaning the height at $t_\\text{end}$ is also by a corresponding fraction of 0.84 units higher.\nWe can verify this visually:\n# animate forward trajectory sol3 = solve(remake(prob,u0=[u0[1]+0.5,u0[2]]),Tsit5(),callback=cb2,saveat=0.01) plt2 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = \u0026quot;t\u0026quot;, legend=:bottomright, color=\u0026quot;black\u0026quot;, xlims=(t0,tend)) hline!(plt2, [0.0], label=false, color=\u0026quot;black\u0026quot;) plot!(plt2, sol3, tspan=(t0,tend), color=[1 2], label = [\u0026quot;z(t)\u0026quot; \u0026quot;v(t)\u0026quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = \u0026quot;t\u0026quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9)) # scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false) list_plots = [] for t in sol3.t tstart = 0.0 plt1 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = \u0026quot;t\u0026quot;, legend=:bottomright, color=\u0026quot;black\u0026quot;) hline!(plt1, [0.0], label=false, color=\u0026quot;black\u0026quot;) plot!(plt1, sol3, tspan=(t0,t), color=[1 2], label = [\u0026quot;z(t)\u0026quot; \u0026quot;v(t)\u0026quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = \u0026quot;t\u0026quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9)) scatter!(plt1,[t,t], sol3(t), color=[1, 2], label=false) plt = plot(plt1,plt2) push!(list_plots, plt) end plot(list_plots[100]) anim = animate(list_plots,every=1)    The original curve is shown in black in the figure above.\nSensitivity analysis with events In more general terms, the last example can be seen as a loss function acting on $z$ at the end time\n$$ L^{\\text{exp}} = z(t_{\\text{end}}), $$\nwhere the superscript \u0026ldquo;exp\u0026rdquo; refers to an explicit definition of the time ($t_{\\text{end}}$) at which we evaluate the loss function. Let $\\alpha$ denote any of the inputs $(z_0,v_0,g,\\gamma)$. The sensitivity with respect to $\\alpha$ is then given by the chain rule\n$$ \\frac{\\text{d}L^{\\text{exp}}}{\\text{d} \\alpha} = \\frac{\\text{d}L^{\\text{exp}}}{\\text{d} z} \\frac{\\text{d}z(t_{\\text{end}})}{\\text{d} \\alpha} = \\frac{\\text{d}z(t_{\\text{end}})}{\\text{d} \\alpha}. $$\nInserting our results for $z(t) = z_{\\rm exp}(t)$ instead of $z(t) = z_{\\rm imp}(t)$ at $t_{\\text{end}}$, a different value for the sensitivity is obtained (a value ignoring the changes in $z$ due to changes in the bouncing time $\\tau$), e.g.,\n$$ \\quad \\frac{\\text{d}L^{\\text{exp}}}{\\text{d} z_0} = \\begin{cases} 1 \u0026amp; \\text{with } z(t) = z_{\\rm exp}(t) ,\\\\\n0.84 \u0026amp; \\text{with } z(t) = z_{\\rm imp}(t) . \\end{cases} $$\nBesides an explicit description of time points, we may also encounter implicitly defined time points. For example, for the bouncing ball with velocity at the impact time $\\tau-$ (i.e., at the left limit) as the quantity of interest,\n$$ L = v(\\tau-), $$\nwe could be interested in the sensitivity of $L$ with respect $g$. Using the velocity $v(t)$ at the explicit time $t = 0.9905$ instead of the implicit $v(\\tau-)$, again a different value for the sensitivity is obtained (a value again ignoring the changes in $\\tau$ and not the one we are looking for), even though $\\tau = 0.9905$ in both cases:\n$$ \\quad \\frac{\\text{d}L}{\\text{d} g} = \\begin{cases} -0.99005 \u0026amp; \\text{with } v(t) = v_{\\rm exp}(t) ,\\\\\n-\\frac{g}{\\sqrt{v_0^2 + 2 g z_0}} = -0.99995 \u0026amp; \\text{ with } v(t) = v_{\\rm imp}(t) . \\end{cases} $$\nThanks to our analytical results for the bouncing ball, the sensitivity computation has been straightforward up to now. However, in most systems, we won\u0026rsquo;t be able to solve analytically a differential equation\n$$ \\text{d}x(t) = f(x,p,t) \\text{d}t $$\nwith initial condition $x_0=x(t_0)$. Instead, we have to numerically solve for the trajectory $x(t)$.\nMore completely, we will in the following derive an adjoint sensitivity method for a loss function\n$$L = \\sum_j L_j(\\tau_j,x(\\tau_j),p) + \\sum_i L^{\\text{exp}}_i(s_j,x(s_i),p) ,$$\nwith $L_i^{\\text{exp}}$ at explicit time points $s_i$, such as $t_{\\text{end}}$, and $L_j(\\tau_j,x(\\tau_j),p)$ at implicit time points, such as $\\tau$, which allows us to compute the sensitivity of $L$ with respect to changes of the parameters or initial condition without the requirement of an analytical solution for $x(t)$.\nBacksolve-Adjoint algorithm for ordinary differential equations Taking derivatives (or finding sensitivities) works in a beautiful mechanical way. We or a computer can find the derivatives of complex expressions by just repeatedly applying the chain rule.\nWe write\n$$\\text{solve}(t_0, x_0, t, p)$$\n$(= x(t))$ for the functional solution of the ODE at time $t$.\nRegarding the computation of the sensitivities (the derivatives of the function solve), we may then choose one of the available algorithms for the given differential equation. Currently, BacksolveAdjoint(), InterpolatingAdjoint(), QuadratureAdjoint(), ReverseDiffAdjoint(), TrackerAdjoint(), and ForwardDiffAdjoint() are compatible with events in ordinary differential equations.\nLet us focus on the BacksolveAdjoint() algorithm which computes the sensitivities\n$$ \\begin{aligned} \\frac{\\text{d}\\thinspace L(\\text{solve}(t_0, x_0, t, p))}{\\text{d}x_{0}} \u0026amp;= \\lambda(t_{0}),\\\\\n\\frac{\\text{d}\\thinspace L(\\text{solve}(t_0, x_0, t, p))}{\\text{d}p} \u0026amp;= \\lambda_{p}(t_{0}), \\end{aligned} $$\nof a loss function $L$ acting on the final state with respect to the initial state and the parameters. It does so by solving an ODE for $\\lambda(s)$ in reverse time from $t$ to $t_0$\n$$ \\begin{aligned} \\frac{\\text{d}\\lambda(s)}{\\text{d}s} \u0026amp;= -\\lambda(s)^\\dagger \\frac{\\text{d} f(\\rightarrow x(s), p, t)}{\\text{d} x(s)} \\\\\n\\frac{\\text{d}\\lambda_{p}(s)}{\\text{d}s} \u0026amp;= -\\lambda(s)^\\dagger \\frac{\\text{d} f(x(s), \\rightarrow p, s)}{\\text{d} p}, \\end{aligned} $$\nwith initial conditions:\n$$ \\begin{aligned} \\lambda(t)\u0026amp;= \\frac{\\text{d}\\thinspace L(\\text{solve}(t_0, x_0, t, p))}{\\text{d}x_{T}}, \\\\\n\\lambda_{p}(t) \u0026amp;= 0. \\end{aligned} $$\nThe arrows ($\\rightarrow$) indicate the variable with respect to which we differentiate, which will become important later when the same variable shows up in multiple function arguments.\nNote that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In BacksolveAdjoint(), we recompute $x(s)$ \u0026ndash; together with the adjoint variables \u0026ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in Chris' MIT 18.337 lecture notes.\nBacksolveAdjoint(), essentially the custom primitive differentiation rule of solve, is the elementary building block needed to derive sensitivities also in more complicated examples:\nConsider a loss depending on the state $x(s_i)$ at fix time points $s_i$ through loss functions $L_i^{\\text{exp}}$,\n$$ L^{\\text{exp}} = \\sum_i L_i^{\\text{exp}}(s_i, x(s_i), p). $$\nWithout contortions we can obtain the sensitivity of $L^{\\text{exp}}$ in $p$ (or in $x_0$) using the tool we have. For those a bit familiar with automatic differentiation, this is perhaps easiest to see if we write $L^{\\text{exp}}$ as pseudo code\nfunction loss(t0, x0, p) x1 = solve(t0, x0, s1, p) L = L1(s1, x1, p) x2 = solve(s1, x1, s2, p) L += L2(s2, x2, p) ... return L end  and consider the problem of automatically differentiating it. You\u0026rsquo;ll just need the primitives of solve, L1 and L2! The sensitivities can be computed by repeated calls to BacksolveAdjoint() on the intervals (s_i, s_{i+1}) backward in time, taking in the sensitivities $\\sum_i L^{\\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation2 as single call BacksolveAdjoint() with discrete callbacks:\n$$ \\begin{aligned} \\frac{\\text{d}\\lambda(t)}{\\text{d}t} \u0026amp;= -\\lambda(t)^\\dagger \\frac{\\text{d} f(\\rightarrow x(t), p, t)}{\\text{d} x(t)} - \\frac{\\text{d} L_i^{\\text{exp}}(\\rightarrow x(t), p)}{\\text{d} x(t)}^\\dagger \\delta(t-s_i), \\\\\n\\frac{\\text{d}\\lambda_{p}(t)}{\\text{d}t} \u0026amp;= -\\lambda(t)^\\dagger \\frac{\\text{d} f(x(t), \\rightarrow p, t)}{\\text{d} p} - \\frac{\\text{d} L_i^{\\text{exp}}( x(t),\\rightarrow p)}{\\text{d} p}^\\dagger \\delta(t-s_i). \\end{aligned} $$\nThe sensitivities of the ordinary solve with respect to the other arguments are also needed and given by\n$$ \\frac{\\text{d}(\\text{solve}(s, x, \\rightarrow t, p))}{\\text{d}t} = f(\\text{solve}(s, x, t, p), p, t) $$\nand\n$$ \\frac{\\text{d}(\\text{solve}(\\rightarrow s, x, t, p))}{\\text{d}s} = -f(x, p, s). $$\nNow we can even properly define the rrule of solve in the sense of DiffRules.jl.\nExplicit events To make BacksolveAdjoint() compatible with explicit events2,\nfunction loss(t0, x0, p) x1- = solve(t0, x0, s1, p) L = L1-(s1, x1-, p) # saved before affect x1 = a(x1-, p, s1) # affect L += L1(s1, x1, p) # saved after affect x2- = solve(s1, x1, s2, p) L += L2-(s2, x2-, p) x2 = a(x2-, p, s2) # affect L += L2(s2, x2, p) ... return L end  we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.5 We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to\n$$ \\begin{aligned} \\lambda({s_j}-) \u0026amp;= \\lambda({s_j})^\\dagger \\frac{\\text{d} a(\\rightarrow x({s_j}-), p, {s_j}-)}{\\text{d} x(s_j-)} \\\\\n\\lambda_p({s_j}-) \u0026amp;= \\lambda_p({s_j}) - \\lambda({s_j})^\\dagger \\frac{\\text{d} a(x({s_j}-), \\rightarrow p, {s_j}-)}{\\text{d} p} \\end{aligned} $$\nwhere $a$ is the affect function applied at the discontinuity. That is, to lift the adjoint from the right to the left limit, we compute a vjp with the adjoint $\\lambda({s_j})$ from the right and the Jacobian of the affect function evaluated immediately before the event time at $s_j-$ .\nIn particular, we apply a loss function callback before and after this update if the state was saved in the forward evolution and entered directly into the loss function.\nImplicit events With implicit events it is similar: Being able to differentiate the ODE when an implicit event terminates the ODE gives us the custom primitive differentiation rule of a solve with implicit callback.\nWe have to account for an important change: besides the value $\\xi = x(\\tau)$ at time of the implicit event, the solver returns the variable event time $\\tau$ itself.\nWe could write for an event condition function $g$\n$$(\\tau_1, \\xi_1) = \\text{solve2}(t_0, x_0, g, p)$$\nto put emphasis on this, or equivalently, compute for a unspecified function $L(t, x, p)$ the result of $\\frac{\\text{d}L}{\\text{d}p}$ with\n$$ \\begin{aligned} \\frac{\\text{d}L}{\\text{d}p} \u0026amp;= \\frac{\\text{d}L( \\text{solve2}(t_0, x_0, g, \\rightarrow p),\\rightarrow p)}{\\text{d}p}\\\\\n\u0026amp;= \\frac{\\text{d}L(\\tau_1({\\color{black}\\rightarrow} p), \\text{solve}(t_0, x_0, \\tau_1({\\color{black}\\rightarrow}p), \\rightarrow p),\\rightarrow p)}{\\text{d}p}, \\end{aligned} $$\nwhich indicates that changing $p$ influences $L$ both through changes in $\\tau_1$ as well as changes in\n$$\\xi_1 = x(\\tau_1-).$$\nThis case where we have a loss function $L = L_1$ depending on $\\tau_1$, $x(\\tau_1)$, and $p$ was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper4.\nTherefore, the sensitivity of the event time with respect to parameters $\\frac{\\text{d}\\tau}{\\text{d}p}$ must be taken into account. Here and in the following we consider only the $p$-dependence of $\\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way4.\nIn a first step, we need to compute the sensitivity of $\\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$. We can apply the implicit function theorem. For this, see that $\\tau_1(p)$ is implicitly defined by $F(p, \\tau_1) = g( \\tau_1, \\text{solve}(t_0, x_0, \\tau_1, p)) = 0$ which yields\n$$ \\begin{aligned} \\frac{\\text{d}\\tau_1(p)}{\\text{d}p} \u0026amp;= - \\left(\\frac{\\text{d}g(\\rightarrow \\tau_1, \\text{solve}(t_0, x_0, \\rightarrow \\tau_1, p))}{\\text{d}\\tau_1}\\right)^{-1} \\frac{\\text{d}g(\\tau_1, \\text{solve}(t_0, x_0, \\tau_1, \\rightarrow p))}{\\text{d}p} .\\\\\n\\end{aligned} $$\nThe total derivative6 inside the bracket is: $$ \\begin{aligned} \\frac{\\text{d}g}{\\text{d}\\tau_1} \\stackrel{\\text{def}}{=} \\frac{\\text{d}g(\\rightarrow \\tau_1, \\text{solve}(t_0, x_0, \\rightarrow \\tau_1, p))}{\\text{d}\\tau_1} \u0026amp;= \\frac{\\text{d}g(\\rightarrow \\tau_1, \\xi_1)}{\\text{d}\\tau_1} + \\frac{\\text{d}g(\\tau_1, \\text{solve}(t_0, x_0, \\rightarrow \\tau_1, p))}{\\text{d}\\tau_1}\\\\\n\\end{aligned} $$\nSince\n$$ \\frac{\\text{d}(\\text{solve}(t_0, x_0, \\rightarrow \\tau_1, p))}{\\text{d}\\tau_1} = f(\\xi_1, p, \\tau_1) $$\nby definition of the ODE, we can write\n$$ \\begin{aligned} \\frac{\\text{d}g(\\tau_1, \\text{solve}(t_0, x_0, \\rightarrow \\tau_1, p))}{\\text{d}\\tau_1} = \\frac{\\text{d}g(\\tau_1, \\xi_1)}{\\text{d} \\xi_1}^{\\dagger} f(\\xi_1, p, \\tau_1). \\end{aligned} $$\nFurthermore, we have $$ \\begin{aligned} \\frac{\\text{d}g(\\tau_1, \\text{solve}(t_0, x_0, \\tau_1, \\rightarrow p))}{\\text{d}p} = \\frac{\\text{d}g(\\tau_1, \\xi_1)}{\\text{d} \\xi_1}^{\\dagger} \\frac{\\text{d}\\text{ solve}(t_0, x_0, \\tau_1,\\rightarrow p)}{\\text{d}p} \\end{aligned} $$ for the second term of $\\dfrac{\\text{d}\\tau_1(p)}{\\text{d}p}$.\nWe can now write the gradient as:\n$$ \\begin{aligned} \\frac{\\text{d}L(\\tau_1({\\color{black}\\rightarrow} p), \\text{solve}(t_0, x_0, \\tau_1({\\color{black}\\rightarrow}p), \\rightarrow p),\\rightarrow p)}{\\text{d}p} \u0026amp;= \\frac{\\text{d}L(\\tau_1(p), \\text{solve}(t_0, x_0, \\tau_1(p), p), \\rightarrow p)}{\\text{d}p} \\\\\n+\u0026amp; \\frac{\\text{d}L(\\tau_1(p), \\text{solve}(t_0, x_0, \\tau_1(p), \\rightarrow p), p)}{\\text{d}p} \\\\\n+\u0026amp; \\frac{\\text{d}L(\\rightarrow \\tau_1(p), \\text{solve}(t_0, x_0, \\rightarrow \\tau_1(p), p), p)}{\\text{d}\\tau_1} \\frac{\\text{d} \\tau_1(p)}{\\text{d}p}, \\end{aligned} $$\nwhich, after insertion of our results above, can be cast into the form:\n$$ \\frac{\\text{d}L}{\\text{d}p} = v^\\dagger \\frac{\\text{d}\\text{ solve}(t_0, x_0, \\tau_1(p), \\rightarrow p)}{\\text{d}p} + \\frac{\\text{d}L(\\tau_1(p), \\text{solve}(t_0, x_0, \\tau_1(p), p), \\rightarrow p)}{\\text{d}p}, $$\nwith\n$$ \\begin{aligned} v \u0026amp;= \\rho \\left(-\\frac{\\text{d}g}{\\text{d}\\tau_1}\\right)^{-1} \\frac{\\text{d}g}{\\text{d}\\xi_1} + \\frac{\\text{d}L(\\tau_1, \\xi_1)}{\\text{d} \\xi_1}, \\end{aligned} $$\nwhere we introduced the scalar pre-factor\n$$ \\begin{aligned} \\rho = \\left( \\frac{\\text{d}L(\\rightarrow \\tau_1, \\xi_1)}{\\text{d}\\tau_1} + \\frac{\\text{d}L(\\tau_1, \\xi_1)}{\\text{d} \\xi_1}^\\dagger f(\\xi_1, p, \\tau_1)\\right). \\end{aligned} $$\nWe have therefore reduced this case to a modification of the original BacksolveAdjoint. This means that if we terminate the ODE integration by an implicit event, we compute the sensitivities as follows:\n  Use an ODE solver to solve forward from the starting value until the event is triggered $$ \\xi_1 = \\text{solve}(t_0, x_0, \\tau_1, p). $$ $(\\tau_1,\\xi_1)$ are the stored values which enter the loss function, which depend on $t_0, x_0$ and $p$.\n  Compute the loss function gradient with respect to the state and event time $$ \\lambda^0_1 = \\frac{\\text{d}L(\\tau_1(p), \\rightarrow \\xi_1, p)}{\\text{d} \\xi_1}, \\quad \\lambda^0_{\\tau_1} = \\frac{\\text{d}L(\\rightarrow \\tau_1(p), \\xi_1, p)}{\\text{d} \\tau_1(p)}. $$\n  (Instead of using the BacksolveAdjoint() algorithm with $\\lambda_1^0$ directly,) use the corrected version containing the dependence on the event time. For this, compute $\\frac{\\text{d}g}{\\text{d}\\tau_1}, \\frac{\\text{d}g}{\\text{d}\\xi_1}$, and $f(\\xi_1, p, \\tau_1)$. Then, the corrected version of the adjoint is given by\n  $$ {\\color{red}\\lambda_1} = - \\left( \\lambda^0_{\\tau_1} + {\\lambda^\\text{0}_1}^\\dagger f(\\xi_1, p, \\tau_1) \\right)\\left(\\frac{\\text{d}g}{\\text{d}\\tau_1}\\right)^{-1} \\frac{\\text{d}g}{\\text{d}\\xi_1} + \\lambda^0_1. $$\nThe correction takes into account a change in the end time and end value of the ODE. ${\\color{red}\\lambda_1}$ can then be used as initial condition to $\\text{backsolve_adjoint}({\\color{red}\\lambda_1}, \\tau_1, \\xi_1, t_0)$ which backpropagates the adjoint ${\\color{red}\\lambda_1}$ at $\\xi_1 = x(\\tau_1-)$ from $\\tau_1$ to $t_0$.\nIf there is an additional affect function $a$ associated with the event, i.e. a right limit, we must additionally compute  $$ \\begin{aligned} {\\lambda_{a,1}^0} = \\frac{\\text{d}L(\\tau_1(p), \\rightarrow a(\\xi_1, p),p)}{\\text{d} a}. \\end{aligned} $$\nCompute the vjp as in the case of a \u0026lsquo;DiscreteCallback\u0026rsquo;  $$ \\lambda_{a,1}^{1} = {\\lambda_{a,1}^0}^\\dagger \\frac{\\text{d}a(\\xi_1,p)}{\\text{d} \\xi_1} $$\nand correct it as above\n$$ \\begin{aligned} {\\color{blue}\\lambda_{a,1}} = - \\left( \\lambda^0_{\\tau_1} + {{\\lambda_{a,1}^1}}^\\dagger f(\\xi_1, p, \\tau_1) \\right)\\left(\\frac{\\text{d}g}{\\text{d}{\\tau_1}}\\right)^{-1} \\frac{\\text{d}g}{\\text{d}{\\xi_1}} + {\\lambda_{a,1}^1}. \\end{aligned} $$\nIf both limits contribute to the loss function, the contributions ${\\color{red}\\lambda_1}$ and ${\\color{blue}\\lambda_{a,1}}$ are added.  Generalization: several events As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle3, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:\n$$ \\text{solve}(t_0, x_0, t, p) = \\text{solve}(s, \\text{solve}(t_0, x_0, s, p), t-s, p), $$\nsuch that also loss function contributions are chained.\n(A good exercise to get familiar with these type of arguments is to verify\n$$\\frac{\\text{d}}{\\text{d}s} \\text{solve}(s, \\text{solve}(t_0, x_0, s, p), t-s, p) = 0.)$$\nEssentially, we know how to address several events already by considering a loss function\nfunction loss2(t0, x0, p) tau1, xi1 = solve2(t0, x0, g, p) L = L1(tau1, xi1, p) tau2, xi2 = solve2(tau1, xi1, g, p) L += L2(tau2, xi2, p) ... return L end  and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for solve2 (rrule for solve2) we derived above.\nDifferentiating by hand, we have the following modification of the method from the previous section, which can be derived choosing a particular function $L$ in the previous section which incorporates the solve from $\\tau_1$ to $t_{\\text{end}}$. Note\n$$ \\lambda_{\\tau_1}^0 = \\frac{\\text{d}(\\text{solve}(\\rightarrow \\tau_1, a(\\xi_1,p), t_{\\text{end}}, p))}{\\text{d} \\tau_1} = - f(a(\\xi_1,p), p, \\tau_1). $$\n Segment the trajectory at the event times. Use $\\text{backsolve_adjoint}(\\lambda^0_\\text{end}, t_\\text{end}, x(t_\\text{end}), \\tau_{J})$ to backprogagate the loss function gradient $\\lambda^0_\\text{end} = \\frac{\\text{d}L(t_\\text{end}, \\rightarrow x(t_\\text{end}), p)}{\\text{d} x(t_\\text{end})}$ from the end state until the right limit of the last event location, obtaining $\\lambda_{J}$ at time $\\tau_{J}$ corresponding to the last event before $t_{\\text{end}}$ with index $J$ (say).\n  Compute ${\\color{red}\\lambda_J}$ as in step 3 (at time $\\tau_{J}$ with $\\lambda^0_J = \\frac{\\text{d}L(\\tau_J(p), \\rightarrow \\xi_J, p)}{\\text{d} \\xi_J}$).\n  As in step 4 compute the vjp (but this time with the sum of the two contributions, $\\lambda_J$ and $\\lambda^0_{a,J} = \\frac{\\text{d}L(\\tau_J(p), \\rightarrow a(\\xi_J,p), p)}{\\text{d} a}$)\n  $$ \\lambda_{a,J}^{1} = \\left({\\lambda_{a,J}^{0} + \\lambda_{J}} \\right)^\\dagger \\frac{\\text{d} a(\\xi_J, p)}{\\text{d} \\xi_J}. $$\n${\\color{blue}\\lambda_{a,J}}$ follows from $\\lambda_{a,J}^{1}$ as in step 5.\nCompute an additional correction term:  $$ {\\color{green}\\lambda_{c,J}} = \\left( \\lambda_J^\\dagger f(a(\\xi_J, p), p, \\tau_J) \\right)\\left(\\frac{\\text{d}g}{\\text{d} \\tau_J}\\right)^{-1} \\frac{\\text{d}g}{\\text{d}{\\xi_J}}, $$\nThe correction has the opposite sign and corresponds to a change in the starting time and starting value in the later time interval ($\\tau_J, t_\\text{end}$) of the ODE.\nBackpropagate $\\lambda_J = {\\color{red}\\lambda_J} + {\\color{blue}\\lambda_{a,J}} + {\\color{green}\\lambda_{c,J}}$ to the next event time $\\tau_{J-1}$ and iterate over the remaining $J-1$ events.  Outlook We are still refining the adjoints in case of implicit discontinuities (ContinuousCallbacks). For further information, the interested reader is encouraged to track the associated issues #383 and #374, and PR #445 in the DiffEqSensitivity.jl package.\nIf you have any questions or comments, please don’t hesitate to contact us ( github.com/frankschae)!\n  Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). \u0026#x21a9;\u0026#xfe0e;\n Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). \u0026#x21a9;\u0026#xfe0e;\n Timo C. Wunderlich and Christian Pehle, Sci. Rep. 11, 12829 (2021). \u0026#x21a9;\u0026#xfe0e;\n Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). \u0026#x21a9;\u0026#xfe0e;\n If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\\lambda_p$. \u0026#x21a9;\u0026#xfe0e;\n For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\\dots,x_N(t))$, the total derivative with respect to the independent variable $t$ is given by the sum of all partial derivatives $$ \\begin{aligned} \\frac{\\text{d}y}{\\text{d}t} \u0026amp;= \\frac{\\text{d}f(\\rightarrow t, x_1(\\rightarrow t),x_2(\\rightarrow t),\\dots,x_N(\\rightarrow t))}{\\text{d}t} \\\\\n\u0026amp;= \\frac{\\text{d}f(\\rightarrow t, x_1(t),x_2(t),\\dots,x_N(t))}{\\text{d}t} + \\frac{\\text{d}f(t, x_1(\\rightarrow t),x_2(t),\\dots,x_N(t))}{\\text{d}t}\\\\\n\u0026amp;+ \\frac{\\text{d}f(t, x_1(t),x_2(\\rightarrow t),\\dots,x_N(t))}{\\text{d}t} + \\dots + \\frac{\\text{d}f(t, x_1(t),x_2(t),\\dots,x_N(\\rightarrow t))}{\\text{d}t}. \\end{aligned} $$ \u0026#x21a9;\u0026#xfe0e;\n   ","date":1626434644,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1626434644,"objectID":"a60e1fb7808c483d902fff1d56e2a3a3","permalink":"https://frankschae.github.io/post/bouncing_ball/","publishdate":"2021-07-16T13:24:04+02:00","relpermalink":"/post/bouncing_ball/","section":"post","summary":"In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.","tags":["GSoC 2021","Hybrid differential equations","Adjoint sensitivity methods","Event handling"],"title":"Sensitivity Analysis of Hybrid Differential Equations","type":"post"},{"authors":[],"categories":[],"content":"In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the \u0026ldquo;butterfly effect\u0026rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.\nSensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control1 2 3. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing4 (LSS) or non-intrusive least squares shadowing5 (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem \u0026ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the DiffEqSensitivity.jl package.\nThe objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:\n$$ \\langle g \\rangle_∞ = \\lim_{T \\rightarrow ∞} \\langle g \\rangle_T, $$ where $$ \\langle g \\rangle_T = \\frac{1}{T} \\int_0^T g(u,s) \\text{d}t. $$ Under the assumption of ergodicity, $\\langle g \\rangle_∞$ only depends on $p$.\nThe Lorenz system One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\\rho$, $\\sigma$, and $\\beta$. Its time evolution is given by the ODE:\n$$ \\begin{pmatrix} \\text{d}x \\\\\n\\text{d}y \\\\\n\\text{d}z \\\\\n\\end{pmatrix} = \\begin{pmatrix} \\sigma (y-x)\\\\\nx(\\rho-z) - y\\\\\nx y - \\beta z \\\\\n\\end{pmatrix}\\text{d}t $$\nFor simplicity, let us fix $\\sigma=10$ and $\\beta=8/3$ and focus only on the sensitivity with respect to $\\rho$. The classic Lorenz attractor is obtained when using $\\rho=28$:\nusing Random; Random.seed!(1234) using OrdinaryDiffEq using Statistics using QuadGK, ForwardDiff, Calculus using DiffEqSensitivity using SparseArrays, LinearAlgebra # simulate 1 trajectory of the Lorenz system forward function lorenz!(du,u,p,t) du[1] = 10*(u[2]-u[1]) du[2] = u[1]*(p[1]-u[3]) - u[2] du[3] = u[1]*u[2] - (8//3)*u[3] end p = [28.0] tspan_init = (0.0,30.0) tspan_attractor = (30.0,50.0) u0 = rand(3) prob_init = ODEProblem(lorenz!,u0,tspan_init,p) sol_init = solve(prob_init,Tsit5()) prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p) sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14) using Plots, LaTeXStrings pl1 = plot(sol_init,vars=(1,2,3), legend=true, label = \u0026quot;initial\u0026quot;, labelfontsize=20, lw = 2, xlabel = L\u0026quot;x\u0026quot;, ylabel = L\u0026quot;y\u0026quot;, zlabel = L\u0026quot;z\u0026quot;, xlims=(-25,30),ylims=(-30,30),zlims=(5,49) ) plot!(pl1, sol_attractor,vars=(1,2,3), label=\u0026quot;attractor\u0026quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49) ) savefig(pl1, \u0026quot;Lorenz_forward.png\u0026quot;)    Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.\nFollowing Refs.4 and 5, we choose\n$$ \\langle z \\rangle_∞ = \\lim_{T \\rightarrow ∞} \\frac{1}{T} \\int_0^T z \\text{d}t $$\nas the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\\rho$.\nfunction compute_objective(sol) quadgk(t-\u0026gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1] end pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L\u0026quot;z(t)\u0026quot;, xlabel=L\u0026quot;t\u0026quot;, label=false, labelfontsize=20,lw = 2) mean_z = [mean(getindex.(sol_attractor.u,3))] int_z = compute_objective(sol_attractor) hline!(pl2, [int_z], label=L\u0026quot;\\langle z\\rangle\u0026quot;, lw = 2) savefig(pl2, \u0026quot;zsingle.png\u0026quot;) # for each value of the parameter, solve 20 times the initial value problem # wrap the procedure inside a function depending on p function Lorenz_solve(p) u0 = rand(3) prob_init = ODEProblem(lorenz!,u0,tspan_init,p) sol_init = solve(prob_init,Tsit5()) prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p) sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14) sol_attractor, prob_attractor end Niter = 10 ps = collect(0.0:1.0:50.0) probs = [] sols = [] zmean = [] zstd = [] for ρ in ps @show ρ ztmp = [] for i=1:Niter sol, prob = Lorenz_solve([ρ]) zbar = compute_objective(sol) push!(sols, sol) push!(probs, prob) push!(ztmp, zbar) end push!(zmean,mean(ztmp)) push!(zstd,std(ztmp)) end pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L\u0026quot;\\langle z\\rangle\u0026quot;, xlabel=L\u0026quot;\\rho\u0026quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50)) savefig(pl3, \u0026quot;zvsrho.png\u0026quot;) pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300)) savefig(pl4, \u0026quot;z.png\u0026quot;)  We obtain:\n  That is, we find a slope of approximately one (almost everywhere except at the kink $\\rho\\approx 23$), and, therefore, we expect a sensitivity of\n$$ \\frac{\\text{d}\\langle z \\rangle_∞}{\\text{d} \\rho} \\approx 1. $$\nConventional forward-mode sensitivity analysis and finite-differencing For non-chaotic systems, we would just use the standard discrete or continuous forward sensitivity methods or even finite-differencing. If we try to compute the sensitivity for the Lorenz system:\nfunction G(p, prob=prob_attractor) tmp_prob = remake(prob,p=p) tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14) res = compute_objective(tmp_sol) @info res res end sense_forward = ForwardDiff.gradient(G,p) sense_calculus = Calculus.gradient(G,p)  we find diverging values:\n$$ \\begin{aligned} \u0026amp; \\frac{\\text{d}\\langle z \\rangle_\\infty}{\\text{d} \\rho} \\Bigg\\rvert_{\\rho=28} \\approx -49899 {\\text{ (ForwardDiff)}} \\\\\n\u0026amp;\\frac{\\text{d}\\langle z \\rangle_\\infty}{\\text{d} \\rho} \\Bigg\\rvert_{\\rho=28} \\approx 472 {\\text{ (Calculus)}} \\end{aligned} $$\nAs pointed out in the NILSS paper, this is because the limit of $T\\rightarrow ∞$ for a fixed initial state does not commute with the differentiation:\n$$ \\frac{\\text{d}}{\\text{d} \\rho} \\langle z \\rangle_∞ \\neq \\lim_{T \\rightarrow ∞} \\frac{\\partial}{\\partial \\rho} \\langle z \\rangle_T $$\nSimilarly, using uncertainty quantification one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:\nprob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p) prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p)) sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01) sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0) list_plots = [] t1 = 0.0 for i in 1:500 t2 = i*0.1 plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true, label = \u0026quot;Float64\u0026quot;, labelfontsize=20, lw = 2, xlabel = L\u0026quot;x\u0026quot;, ylabel = L\u0026quot;y\u0026quot;, zlabel = L\u0026quot;z\u0026quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)) plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=\u0026quot;Float32\u0026quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)) push!(list_plots, plt1) end anim = animate(list_plots,every=1) pl1 = plot(sol1,vars=(1,2,3), legend=true, label = \u0026quot;Float64\u0026quot;, labelfontsize=20, lw = 2, xlabel = L\u0026quot;x\u0026quot;, ylabel = L\u0026quot;y\u0026quot;, zlabel = L\u0026quot;z\u0026quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48) ) plot!(pl1, sol2,vars=(1,2,3), label=\u0026quot;Float32\u0026quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48) ) savefig(pl1, \u0026quot;Lorenz_Floats.png\u0026quot;)    Without animation:\n  Luckily, the shadowing lemma states:\n Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (\u0026ldquo;shadows\u0026rdquo;) the numerically computed one.\n Shadowing methods The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions. That implies that we aim at finding two trajectories, one with $p$ and one with $p+\\delta p$, which do not diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. 4 and 5, including a visualization of both effects in Fig. 1 of Ref. 5.\nLSS and NILSS for the Lorenz system Switching to LSS or NILSS within the SciML ecosystem is straightforward by either defining the associated LSS (ForwardLSSProblem or AdjointLSSProblem) or NILSS problem (NILSSProblem) type manually:\n# objective g(u,p,t) = u[end] #### # LSS #### lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g) @show shadow_forward(lss_problem) # 1.0095888187322035 lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g) @show shadow_forward(lss_problem) # 1.0343951385924328 lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g) @show shadow_forward(lss_problem) # 1.0284286902740765 adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g) @show shadow_adjoint(adjointlss_problem) # 1.028428690274077  or by setting the sensealg= kwarg in solve():\n# select via sensealg in solve using Zygote function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing) _prob = remake(prob_attractor,p=p) _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g) sum(getindex.(_sol.u,3)) end dp1 = Zygote.gradient((p)-\u0026gt;GLSS(p),p) # 0.9694728321500617  Note that we have implemented three different options for forward shadowing with LSS():\n CosWindowing() (default) Cos2Windowing() time dilation with a factor of $\\alpha$.  Additionally, an adjoint implementation AdjointLSS() is available that is particularly recommended for a large number of system parameters. Based on the values computed above, we can easily check that AdjointLSS(alpha=10.0) agrees perfectly with ForwardLSS(alpha=10.0). In all cases considered, we find the expected sensitivity value of $\\approx 1$.\nHowever, the use of LSS() is (typically) much more expensive than the use of NILSS(), because LSS() needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large. The computational and memory costs of NILSS() scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making NILSS() more efficient.\nIn the NILSS() algorithm, the user can control the number of steps per segment as well as the number of segments.\n#### # NILSS #### # make sure trajectory is fully on the attractor Random.seed!(1234) tspan_init = (0.0,100.0) tspan_attractor = (100.0,120.0) u0 = rand(3) prob_init = ODEProblem(lorenz!,u0,tspan_init,p) sol_init = solve(prob_init,Tsit5()) prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p) nseg = 100 # number of segments on time interval nstep = 2001 # number of steps on each segment nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g) @show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089  If the number of segments is chosen too small, a warning is thrown:\nnseg = 20 # number of segments on time interval nstep = 2001 # number of steps on each segment nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g) @show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789 # Warning: Detected a large value of ξ at the beginning of a segment. # └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474  In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper5.\nOutlook With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as\n NILSAS6 FD-NILSS7  in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding issue in the DiffEqSensitivity.jl package.\nIf you have any questions or comments, please don’t hesitate to contact me!\n  Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. 1, 035009 (2020). \u0026#x21a9;\u0026#xfe0e;\n Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. 2, 035004 (2021). \u0026#x21a9;\u0026#xfe0e;\n Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). \u0026#x21a9;\u0026#xfe0e;\n Qiqi Wang, Rui Hu, et al. J. Comput. Phys 26, 210-224 (2014) \u0026#x21a9;\u0026#xfe0e;\n Angxiu Ni and Qiqi Wang. J. Comput. Phys 347, 56-77 (2017). \u0026#x21a9;\u0026#xfe0e;\n Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys 395, 690-709, (2019) \u0026#x21a9;\u0026#xfe0e;\n Angxiu Ni, Qiqi Wang et al., J. Comput. Phys 394, 615-631 (2019) \u0026#x21a9;\u0026#xfe0e;\n   ","date":1625216902,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1625216902,"objectID":"263091e46f2c8aead98407a523d8a2fe","permalink":"https://frankschae.github.io/post/shadowing/","publishdate":"2021-07-02T11:08:22+02:00","relpermalink":"/post/shadowing/","section":"post","summary":"In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.","tags":["GSoC 2021","julia","Adjoint sensitivity methods","Forward sensitivity methods","Shadowing","Chaotic systems"],"title":"Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems","type":"post"},{"authors":[],"categories":[],"content":"I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek. This year\u0026rsquo;s project is embedded within the NumFocus/ SciML organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.\nThis first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).\nNote: Please check the SciMLSensitivity.jl docs for a maintained neural hybrid DE tutorial!\nHybrid Differential Equations DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by callbacks. Evidently, the incorporation of discontinuities allows a user to specify changes (events) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by DiscreteCallbacks, implicit events have to be specified by ContinuousCallbacks. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to 0. Thus, implicit events require some sort of rootfinding procedure.\nSome relevant examples for hybrid DEs with discrete or continuous callbacks are:\n quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.1 (ContinuousCallback). a bouncing ball2 (ContinuousCallback). classical point process models, such as a Poisson process3. digital controllers4, where a continuous system dynamics is controlled by a discrete-time controller (DiscreteCallback). pharmacokinetic models5, where explicit dosing times change the drug concentration in the blood (DiscreteCallback). The simplest possible example being the one-compartment model. kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (DiscreteCallback).  The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. 2 and 3.\nKicked harmonic oscillator Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form\n$$ \\ddot{x}(t) + a\\cdot\\dot{x}(t) + b \\cdot x(t) = 0 , $$\nwhere $a=0.1$ and $b=1$ with initial conditions\n$$ \\begin{aligned} x(t=0) \u0026amp;= 1 \\\\\nv(t=0) \u0026amp;= \\dot{x}(t=0) = 0. \\end{aligned}\n$$\nThis second-order ODE can be reduced to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the DifferentialEquations.jl package. (Instead of doing this reduction manually, we could also use ModelingToolkit.jl to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a SecondOrderODEProblem implemented.) The Julia code reads:\nusing DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity using Zygote using Random u0 = Float32[1.; 0.] tspan = (0.0f0,50.0f0) dtsave = 0.5f0 t = tspan[1]:dtsave:tspan[2] function oscillator!(du,u,p,t) du[1] = u[2] du[2] = - u[1] - 1//10*u[2] return nothing end prob_data = ODEProblem(oscillator!,u0,tspan) # ODE without kicks pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[\u0026quot;x(t)\u0026quot; \u0026quot;v(t)\u0026quot;])    We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as 1.\nkicktimes = tspan[1]:1:tspan[2] function kick!(integrator) integrator.u[end] += one(eltype(integrator.u)) end cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false)) sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t) t_data = sol_data.t ode_data = Array(sol_data) # visualize data pl1 = plot(t_data,ode_data[1,:],label=\u0026quot;data x(t)\u0026quot;) plot!(pl1,t_data,ode_data[2,:],label=\u0026quot;data v(t)\u0026quot;) pl2 = plot(t_data[1:20],ode_data[1,1:20],label=\u0026quot;data x(t)\u0026quot;) plot!(pl2,t_data[1:20],ode_data[2,1:20],label=\u0026quot;data v(t)\u0026quot;) pl = plot(pl2, pl1, layout=(1,2), xlabel=\u0026quot;t\u0026quot;)     The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting save_positions=(true,true), the kicks would be saved before and after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations6, we now aim at learning (potentially) missing parts of the model from these data traces.\nHigh domain knowledge For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the affect! function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of integrator.u. If we further set the initial value of that component to 1, then the neural network only has to learn that du[3] is 0. In other words, the output of the neural network must be 0 for all states u.\nRandom.seed!(123) nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1)) p_nn1 = initial_params(nn1) function f1!(du,u,p,t) du[1] = u[2] du[2] = - u[1] - 1//10*u[2] du[3] = nn1(u[1:2], p)[1] return nothing end affect!(integrator) = integrator.u[2] += integrator.u[3] cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false)) z0 = Float32[u0;one(u0[1])] prob1 = ODEProblem(f1!,z0,tspan,p_nn1)  We can easily compare the time evolution of the neural hybrid DE with respect to the data:\n# to visualize the predictions of the trained neural network below function visualize(prob,p) _prob = remake(prob,p=p) ode_pred = Array(solve(_prob,Tsit5(),callback=cb, saveat=dtsave))[1:2,:] pl1 = plot(t_data,ode_pred[1,:],label=\u0026quot;x(t)\u0026quot;) scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=\u0026quot;data x(t)\u0026quot;) pl2 = plot(t_data,ode_pred[2,:],label=\u0026quot;v(t)\u0026quot;) scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=\u0026quot;data v(t)\u0026quot;) pl = plot(pl1, pl2, layout=(1,2), xlabel=\u0026quot;t\u0026quot;) return pl, sum(abs2,ode_data .- ode_pred) end pl = plot(solve(prob1,Tsit5(),saveat=t, callback=cb ),label=[\u0026quot;x(t)\u0026quot; \u0026quot;v(t)\u0026quot; \u0026quot;u3(t)\u0026quot;])    which (of course) doesn\u0026rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:\n### loss function function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint()) _prob = remake(prob,p=p) pred = Array(solve(_prob,Tsit5(),callback=cb, saveat=dtsave,sensealg=sensealg))[1:2,:] sum(abs2,ode_data .- pred) end loss(p_nn1)  The recently implemented tools are deeply hidden within the DiffEqSensitivity.jl package. However, while the user could previously only choose discrete sensitivities such as ReverseDiffAdjoint() or ForwardDiffAdjoint() that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as BacksolveAdjoint(), InterpolatingAdjoint(), and QuadratureAdjoint() as the sensealg for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., Chris' talk at the SciML symposium at SIAM CSE.\n################################### # training loop # optimize the parameters for a few epochs with ADAM function train(prob, p_nn; sensealg=BacksolveAdjoint()) opt = ADAM(0.0003f0) list_plots = [] losses = [] for epoch in 1:200 println(\u0026quot;epoch: $epoch / 200\u0026quot;) _dy, back = Zygote.pullback(p -\u0026gt; loss(p, prob=prob, sensealg=sensealg), p_nn) gs = @time back(one(_dy))[1] push!(losses, _dy) if epoch % 10 == 0 # plot every xth epoch pl, test_loss = visualize(prob, p_nn) println(\u0026quot;Loss (epoch: $epoch): $test_loss\u0026quot;) display(pl) push!(list_plots, pl) end Flux.Optimise.update!(opt, p_nn, gs) println(\u0026quot;\u0026quot;) end return losses, list_plots end # plot training loss losses, list_plots = train(prob1, p_nn1) pl1 = plot(losses, lw = 1.5, xlabel = \u0026quot;epoch\u0026quot;, ylabel=\u0026quot;loss\u0026quot;, legend=false) pl2 = list_plots[end] pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t, callback=cb ), label=[\u0026quot;x(t)\u0026quot; \u0026quot;v(t)\u0026quot; \u0026quot;u3(t)\u0026quot;]) pl = plot(pl2,pl3)    We see the expected constant value of u[3], indicating a kick to the velocity of +=1, at the kicking times over the full time interval.\nReducing the domain knowledge If less physical information is included in the model design, the training becomes more difficult, e.g., due to local minima. Possible modification for the kicked oscillator could be\n changing the initial condition of the third component of u, using another affect function affect!(integrator) = integrator.u[2] = integrator.u[3], dropping the knowledge that only u[2] gets a kick by using a neural network with 2 outputs (+ a fourth component in the ODE):  affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4] function f2!(du,u,p,t) du[1] = u[2] du[2] = - u[1] - 1//10*u[2] du[3:4] .= nn2(u[1:2], nn_weights) return nothing end   fitting the parameters $a$ and $b$ simultaneously:  function f3!(du,u,p,t) a = p[end-1] b = p[end] nn_weights = p[1:end-2] du[1] = u[2] du[2] = -b*u[1] - a*u[2] du[3:4] .= nn2(u[1:2], nn_weights) return nothing end   inferring the entire underlying dynamics using a neural network with 4 outputs:  function f4!(du,u,p,t) Ω = nn3(u[1:2], p) du[1] = Ω[1] du[2] = Ω[2] du[3:4] .= Ω[3:4] return nothing end   etc.  Outlook With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to\n refine the adjoints in case of implicit discontinuities (ContinuousCallbacks) and support direct usage through the jump problem interface  in the upcoming weeks. For further information, the interested reader is encouraged to look at the open issues in the DiffEqSensitivity.jl package.\nIf you have any questions or comments, please don’t hesitate to contact me!\n  Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. 2, 035004 (2021) \u0026#x21a9;\u0026#xfe0e;\n Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). \u0026#x21a9;\u0026#xfe0e;\n Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). \u0026#x21a9;\u0026#xfe0e;\n Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). \u0026#x21a9;\u0026#xfe0e;\n Chris Rackauckas, Yingbo Ma, et al., \u0026ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.\u0026rdquo; (2020). \u0026#x21a9;\u0026#xfe0e;\n Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). \u0026#x21a9;\u0026#xfe0e;\n   ","date":1623847817,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1623847817,"objectID":"c5c641840c6146609c2e90c018c85a3f","permalink":"https://frankschae.github.io/post/hybridde/","publishdate":"2021-06-16T14:50:17+02:00","relpermalink":"/post/hybridde/","section":"post","summary":"I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.","tags":["GSoC 2021","julia","Hybrid differential equations","Adjoint sensitivity methods","Event handling"],"title":"Neural Hybrid Differential Equations","type":"post"},{"authors":[],"categories":[],"content":"","date":1615332097,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615332097,"objectID":"6b8484ee245fb6a73711cfc61c673627","permalink":"https://frankschae.github.io/software/sciml/","publishdate":"2021-03-10T00:21:37+01:00","relpermalink":"/software/sciml/","section":"software","summary":"Contributions to the SciML ecosystem in Julia, especially the DiffEqSensitivity.jl package for sensitivity analysis utilities, the StochasticDiffEq.jl package for stochastic differential equations solvers, and the DiffEqNoiseProcess package for tools to develop noise processes for differential equations.","tags":[],"title":"SciML Scientific Machine Learning Software","type":"software"},{"authors":[],"categories":[],"content":"","date":1615332037,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615332037,"objectID":"7d136c46f74da06acca4c2ac00a7b219","permalink":"https://frankschae.github.io/software/mitosisstochasticdiffeq/","publishdate":"2021-03-10T00:20:37+01:00","relpermalink":"/software/mitosisstochasticdiffeq/","section":"software","summary":"Implementation of the backward filter and the forward change of measure of the Automatic Backward Filtering Forward Guiding paradigm. Joint work with [Moritz Schauer](https://github.com/mschauer).","tags":[],"title":"MitosisStochasticDiffEq.jl","type":"software"},{"authors":["FS in collaboration with Pavel Sekatski","Martin Koppenhöfer","Niels Lörch","Christoph Bruder","and Michal Kloc"],"categories":[],"content":"Conceptually, it is straightforward to determine the time evolution of a quantum system for a fixed initial state given its (time-dependent) Hamiltonian or Lindbladian. Depending on the physical context, the dynamics is described by an ordinary or stochastic differential equation. In quantum state control, which is of paramount importance for quantum computation, we aim at solving the inverse problem. That is, starting from a distribution of initial states, we seek protocols that allow us to reach a desired target state by optimization of free parameters of the differential equation (control drives) in a certain time interval. To solve this control problem, we implement the system dynamics as part of a fully differentiable program and use a loss function that quantifies the distance from the target state. Specifically, we employ a neural network that maps an observation of the state of the qubit to a control drive defined via the differential equation for each time interval. To implement efficient training, we backpropagate the gradient information from the loss function through the SDE solver using adjoint sensitivity methods. Such a procedure should ultimately combine powerful tools from machine learning and scientific computation.\n","date":1615332029,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615332029,"objectID":"0dda26cfe2920348a8f4a2cf6ef7ac1d","permalink":"https://frankschae.github.io/project/dp_for_control/","publishdate":"2021-03-10T00:20:29+01:00","relpermalink":"/project/dp_for_control/","section":"project","summary":"Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.","tags":["SciML","differentiable programming","NODE","NSDE","adjoint sensitivity methods","automatic differentiation","quantum control"],"title":"Control of (Stochastic) Quantum Dynamics with Differentiable Programming","type":"project"},{"authors":["FS in collaboration with Eliska Greplova","Tomoki Hirosawa","Jozef Bucko","František Herman","Rebekka Garreis","Chuyao Tong","Annika Kurzmann","Thomas Ihn","Hideaki Maebashi","Hiroyasu Matsuura","and Masao Ogata"],"categories":[],"content":"","date":1615330603,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615330603,"objectID":"94d4899d49aece16c0d03e7ab9c4963f","permalink":"https://frankschae.github.io/project/reconstruction/","publishdate":"2021-03-09T23:56:43+01:00","relpermalink":"/project/reconstruction/","section":"project","summary":"We address the challenges of connecting measurements of a physical system to an underlying theoretical description.","tags":[],"title":"Automated Data-driven Reconstruction of Physical Properties","type":"project"},{"authors":["FS in collaboration with Julian Arnold","Flemming Holtorf","Eliska Greplova","Agnes Valenti","Martin Zonda","Axel Lode","Gregor Boschung","Sebastian Huber","and Niels Lörch"],"categories":[],"content":"Artificial neural networks have successfully been used to identify phase transitions from data and classify data into distinct phases in an automated fashion. The power and success of these approaches (e.g., \u0026ldquo;learning by confusion\u0026rdquo; or the \u0026ldquo;prediction-based method\u0026rdquo;) can be attributed to the ability of deep neural networks to learn arbitrary functions. However, the larger a neural network, the more computational resources are needed to train it, and the more difficult it is to understand its decision making. In this project, we establish a solid theoretical foundation of popular machine-learning methods that rely on neural networks for detecting phase transitions and develop new efficient numerical routines to identify phase transitions directly from data. In the future, our methods could make it possible to detect as yet unknown phase transitions, for instance in quantum simulators or in novel materials.\n","date":1615330603,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615330603,"objectID":"96c9e3dc2a1485e968cc89f130ac8af0","permalink":"https://frankschae.github.io/project/ml_for_pt/","publishdate":"2021-03-09T23:56:43+01:00","relpermalink":"/project/ml_for_pt/","section":"project","summary":"Data-driven methods based on sample instances of the state of a physical system as a function of the system's parameters.","tags":["Machine learning","analytical predictors","neural networks","unsupervised learning"],"title":"Machine Learning for Phase Transitions","type":"project"},{"authors":["FS in collaboration with Laurent de Forges de Parny","Miguel Bastarrachea","Axel Lode","and Andreas Buchleitner"],"categories":[],"content":"We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior of the particles launched either at the single-particle ground state or saddle-point energy, in a time-independent potential. We complement these results by a characterization of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time evolution of a single particle’s von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X)—which also allows us to extrapolate our results for increasing particle numbers.\n","date":1615282751,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615282751,"objectID":"4b16c3ed4d0b9e975df9b36f75807256","permalink":"https://frankschae.github.io/project/mbdynamics/","publishdate":"2021-03-09T10:39:11+01:00","relpermalink":"/project/mbdynamics/","section":"project","summary":"Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.","tags":["Many-body physics","exact diagonalization","Bose-Hubbard model","MCTDH-X"],"title":"Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well ","type":"project"},{"authors":[],"categories":[],"content":"Project summary In this project, we have implemented new promising tools within the SciML organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations. The high weak order solvers will allow for massive performance advantages for fitting expectations of equations. Instead of automatic differentiation (AD) through the operations of an SDE solver, which scales poorly in memory, one can now use efficient stochastic adjoint sensitivity methods.\nBlog posts The following posts describe the work during the entire period in more detail:\n  GSoC 2020: High weak order SDE solvers and their utility in neural SDEs  High weak order SDE solvers  Docs The documentation of the solvers is available here. Docs with respect to the adjoint sensitivity tools will be available here.\nAchievements Please find below a list of the PRs carried out during GSoC in the different repositories in chronological order within the SciML ecosystem.\nStochasticDiffEq.jl Merged:\n  Inplace version of DRI1 scheme  RI1 method  tstop fixes for reverse time propagation  Fixes for EulerHeun scheme  Speed up the tests for the DRI1 and RI1 schemes  RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes  RDI1WM scheme  Adaptive version of the stochastic Runge-Kutta schemes by embedding  RS1 and RS2 schemes  PL1WM scheme  NON scheme  Citations for weak methods  Stochastic improved and modified Euler methods  COM scheme  Computationally more efficient NON variant (NON2)  Open:\n  Static array tests  Levy area for non-commutative noise processes  DiffEqSensitivity.jl Merged:\n  Adjoint sensitivities for steady states  Concrete_solve dispatch for steady state problem  BacksolveAdjoint for SDEs  GPU savety for SDE BacksolveAdjoint  Tests for concrete solve with respect to SDEs  Alternative differentiation choices (vjps) for noise Jacobian  Fixes and tests for inplace formulation of BacksolveAdjoint  Efficient SDE BacksolveAdjoint for scalar noise  Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms  InterpolatingAdjoint for SDEs  Citations for backsolve, steadystate and interpolation adjoint  Allow for more general noise processes: replace NoiseGrid by NoiseWrapper  Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs  Cheaper non-diagonal noise tests  Open:\n  Support adjoints for SDEs written in the Ito sense  DiffEqNoiseProcess.jl Merged:\n  Multi-dimensional Brownian motion tests  Bug fix for inplace form of NoiseGrid  Reversible NoiseWrapper  Relax the size constraints of the available noise processes  Generalization of the real-valued white noise process function  Fix of an extraction issue with NoiseGrid  Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory  Extraction and endpoint fixes on NoiseWrapper  Reversal of SDEs written in the Ito sense  DiffEqGPU.jl Merged:\n  Memory efficient reduction function for ensemble problems  ModelingToolkit.jl Merged:\n  modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense  DiffEqDevTools.jl Merged:\n  NoiseWrapper alternative for analyticless convergence tests of SDE solvers  test_convergence() dispatch for ensemble simulations  Work precision set for ensemble problems  DiffEqBase.jl Merged:\n  Fix concrete_solve tests  Future work There is still a lot that we\u0026rsquo;d like to do, e.g.,\n Writing up more docs and examples Implementing drift-implicit weak stochastic Runge-Kutta solvers Finishing the SDE adjoints for the Ito sense Implementing a virtual Brownian tree to store the noise processes in O(1) memory Setting up an OptimalControl library that allows for easy usage of the new tools within a symbolic interface Benchmarking of the new solvers and adjoints  Contributions, suggestions \u0026amp; comments are always welcome! You might like to join our slac channels #diffeq-bridged and #neuralsde to get in touch.\nAcknowledgement I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback. I would also like to thank Christoph Bruder, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts. Many thanks to the very supportive julia community and to Google\u0026rsquo;s open source program for funding this experience!\n","date":1598475589,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1598475589,"objectID":"c74c1755693bb9cf28d7bf9019cfb1ee","permalink":"https://frankschae.github.io/post/gsoc-2020/","publishdate":"2020-08-26T22:59:49+02:00","relpermalink":"/post/gsoc-2020/","section":"post","summary":"Project summary In this project, we have implemented new promising tools within the SciML organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.","tags":["GSoC 2020"],"title":"High weak order solvers and adjoint sensitivity analysis for stochastic differential equations","type":"post"},{"authors":[],"categories":[],"content":"This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.\nAfter an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process. Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics. Finally, we discuss how to run simulations on GPU hardware.\nThroughout this post, we shall use the vector notation $X(t)$ to denote the solution of the d-dimensional Ito SDE system\n$$ dX(t) = a(t,X(t)) dt + b(t,X(t)) dW $$\nwith an m-dimensional driving Wiener process W(t) in the time span $\\mathbb{I}=[t_0, T]$, where $a: \\mathbb{I}\\times\\mathbb{R}^d \\rightarrow \\mathbb{R}^d$ and $b: \\mathbb{I}\\times \\mathbb{R}^{d} \\rightarrow \\mathbb{R}^{d \\times m}$ are continuous functions which fulfill a global Lipschitz condition.1 For simplicity, we write $X(t)$ for both time-discrete approximations and continuous-time random variables in the following.\nStrong convergence Suppose that we encounter the following problem: Given noisy observations $Z(t)$ (e.g., originating from measurement noise), what is the best estimate $\\hat{X}(t)$ of a stochastic system $X(t)$ satisfying the form above. Intuitively, we aim at filtering away the noise from the observations in an optimal way. Such tasks are well known as filtering problems.\nTo solve a filtering problem, we need a solver whose sample paths $Y(t)$ are close to the ones of the stochastic process $X(t)$, i.e., the solver should allow us to reconstruct correctly the numerical solution of each single trajectory of an SDE.\nIntroducing the absolute error at the final time $T$ as $$ \\rm{E}(|X(T) -Y(T)|) \\leq \\sqrt{\\rm{E}(|X(T)-Y(T)|^2)}, $$ we define convergence in the strong sense with order $p$ of a time discrete approximation $Y(T)$ with step size $h$ to the solution $X(T)$ of a SDE at time $T$ if there exists a constant $C$ (independent of $h$) and a constant $\\delta \u0026gt; 0$ such that $$ \\rm{E}(|X(T) -Y(T)|) \\leq C \\cdot h^p, $$ for each $h \\in [0, \\delta]$.\nThe StochasticDiffEq package contains various state-of-the-art solvers for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms. For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI) can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where b has only entries on its diagonal or $m=1$. The main difficulty for the construction of strong methods with an order \u0026gt; 1/2 arises from the need of an accurate estimation of multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of dW in the case of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed in the case of non-commutative noise processes.2 Currently, we are also implementing those iterated integrals in the StochasticDiffEq library.\nWeak convergence Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the expected value of the solution in many situations. In these cases, methods for the weak approximation are sufficient and \u0026ndash; due to the less restrictive formulation of the objective \u0026ndash; those solvers are computationally cheaper than their strong counterparts. For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics. Our new contributions are particularly appealing for many-body simulations, which are the computationally most demanding problems in quantum mechanics.\nWe define convergence in the weak sense with order $p$ of a time-discrete approximation $Y(T)$ with step size $h$ to the solution $X(T)$ of a SDE at time $T$ if there exists a constant $C$ (independent of $h$) and a constant $\\delta \u0026gt; 0$ such that $$ |\\rm{E}(g(X(T))) -\\rm{E}(g(Y(T)))| \\leq C \\cdot h^p, $$ $~$\nfor any polynomial $g$ for each $h \\in [0, \\delta]$.\nWe demonstrate below that high weak order solvers are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.\nNew high weak order methods A list of all new weak solvers is available in the SciML documentation. Note that we also implemented methods designed for the Stratonovich sense. For the subsequent examples regarding Ito SDEs, we use only a subset of the plethora of second-order weak solvers. We employ the DRI1()3, RD1WM()4, and RD2WM()4 methods due to Debrabant \u0026amp; Rößler and Platen\u0026rsquo;s PL1WM()1 method. We compare those methods to the strong Euler-Maruyama EM()1 and the simplified Euler-Maruyama SimplifiedEM()1 schemes. The latter is the simplest weak solver, where the Gaussian increments of the strong Euler-Maruyama scheme are replaced by two-point distributed random variables with similar moment properties.\nRößler\u0026rsquo;s SRK schemes are particularly designed to scale well with the number of Wiener processes m, since only 2m-1 random variables have to be drawn and since the number of function evaluations for the drift and the diffusion terms is independent of m. PL1WM() in contrast needs to simulate m(m+1)/2 random variables but a smaller number of order conditions needs to be fulfilled.\nConvergence tests As in the first blog post, let us consider the multi-dimensional SDE with non-commuting noise3:\n$$ \\scriptstyle d \\begin{pmatrix} X_1 \\\\ X_2 \\end{pmatrix} = \\begin{pmatrix} -\\frac{273}{512} \u0026amp; \\phantom{X_2}0 \\\\ -\\frac{1}{160} \\phantom{X_2} \u0026amp; -\\frac{785}{512}+\\frac{\\sqrt{2}}{8} \\end{pmatrix} \\begin{pmatrix} X_1 \\\\ X_2 \\end{pmatrix} dt + \\begin{pmatrix} \\frac{1}{4} X_1 \u0026amp; \\frac{1}{16} X_1 \\\\ \\frac{1-2\\sqrt{2}}{4} X_2 \u0026amp; \\frac{1}{10}X_1 +\\frac{1}{16} X_2 \\end{pmatrix} d \\begin{pmatrix} W_1 \\\\ W_2 \\end{pmatrix} $$\nwith initial value $$ ~$$\n$$ X(t=0)= \\begin{pmatrix} 1 \\\\ 1\\end{pmatrix},$$\nwhere the expected value of the solution can be computed analytically\n$$ \\rm{E}\\left[ f(X(t)) \\right] = \\exp(-t),$$\nfor the function $f(x)=(x_1)^2$, which we use to test the weak convergence order of the algorithms in the following.\nTo compute the expected value numerically, we sample an ensemble of numtraj = 1e6 trajectories for different step sizes dt.\nusing StochasticDiffEq using Test using Random using Plots using DiffEqDevTools function prob_func(prob, i, repeat) remake(prob,seed=seeds[i]) end u₀ = [1.0,1.0] function f1!(du,u,p,t) @inbounds begin du[1] = -273//512*u[1] du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2] end return nothing end function g1!(du,u,p,t) @inbounds begin du[1,1] = 1//4*u[1] du[1,2] = 1//16*u[1] du[2,1] = (1-2*sqrt(2))/4*u[1] du[2,2] = 1//10*u[1]+1//16*u[2] end return nothing end dts = 1 .//2 .^(3: -1:0) tspan = (0.0,3.0) h2(z) = z^2 # but apply it only to u[1] prob = SDEProblem(f1!,g1!,u₀,tspan,noise_rate_prototype=zeros(2,2)) numtraj = Int(1e6) seed = 100 Random.seed!(seed) seeds = rand(UInt, numtraj) ensemble_prob = EnsembleProblem(prob; output_func = (sol,i) -\u0026gt; (h2(sol[end][1]),false), prob_func = prob_func ) sim = test_convergence(dts,ensemble_prob,DRI1(), save_everystep=false,trajectories=numtraj,save_start=false,adaptive=false, weak_timeseries_errors=false,weak_dense_errors=false, expected_value=exp(-3.0) )  The object sim defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the DRI1() scheme. Repeating this call to the test_convergence() function for the other aforementioned solvers, we obtain the convergence plot:\n  Note that the SimplifiedEM and the EM scheme fall on top of each other. DRI1() achieves the smallest errors for a fixed dt in this study.\nWork-Precision Diagrams Ultimately, we are not only interested in the general convergence slope of an algorithm but also in its speed. We\u0026rsquo;d like to select the fastest method depending on the permitted tolerance. This is commonly studied using work-precision diagram. Thanks to new routines, a user can generate a work-precision diagram by the following code\nreltols = 1.0 ./ 4.0 .^ (1:4) abstols = reltols#[0.0 for i in eachindex(reltols)] setups = [ Dict(:alg=\u0026gt;DRI1(),:dts=\u0026gt;dts,:adaptive=\u0026gt;false), Dict(:alg=\u0026gt;PL1WM(),:dts=\u0026gt;dts,:adaptive=\u0026gt;false), Dict(:alg=\u0026gt;EM(),:dts=\u0026gt;dts,:adaptive=\u0026gt;false), Dict(:alg=\u0026gt;SimplifiedEM(),:dts=\u0026gt;dts,:adaptive=\u0026gt;false), Dict(:alg=\u0026gt;RDI2WM(),:dts=\u0026gt;dts,:adaptive=\u0026gt;false), Dict(:alg=\u0026gt;RDI1WM(),:dts=\u0026gt;dts,:adaptive=\u0026gt;false) ] test_dt = 1//10000 appxsol_setup = Dict(:alg=\u0026gt;EM(), :dt=\u0026gt;test_dt) wp = @time WorkPrecisionSet(ensemble_prob, abstols,reltols,setups,test_dt; maxiters = 1e7,verbose=false, save_everystep=false,save_start=false, appxsol_setup = appxsol_setup, expected_value=exp(-3.0), trajectories=numtraj, error_estimate=:weak_final) plt = plot(wp;legend=:bottomleft)    Therefore, DRI1 has the best performance in this non-commutative noise case if the error is supposed to stay below 1e-3. For larger permitted errors, the SimplifiedEM scheme might be a good choice. However, the first order methods are outclassed when high precision is more of a concern. We plan to perform more in-depth benchmarks in the near future what will be reported on the SciML news. Stay tuned!\nAdaptive step-size control Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.4 The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.\nTo use adaptive step-size control, it is sufficient to set adaptive=true (default setting). Optionally, one may also pass absolute and relative tolerances.\nThe following julia code simulates the stochastic version of the Brusselator equations with intitial condition\n$$ X(t=0)= \\begin{pmatrix} 0.1 \\\\ 0\\end{pmatrix},$$\non a time span $\\mathbb{I}=[0, 100]$ for adaptive (sol) and fixed step sizes (sol_na):\nusing StochasticDiffEq, DiffEqNoiseProcess, Random using Plots using DiffEqGPU function prob_func(prob, i, repeat) Random.seed!(seeds[i]) W = WienerProcess(0.0,0.0,0.0) remake(prob,noise=W) end function brusselator_f!(du,u,p,t) @inbounds begin du[1] = (p[1]-1)*u[1]+p[1]*u[1]^2+(u[1]+1)^2*u[2] du[2] = -p[1]*u[1]-p[1]*u[1]^2-(u[1]+1)^2*u[2] end nothing end function scalar_noise!(du,u,p,t) @inbounds begin du[1] = p[2]*u[1]*(1+u[1]) du[2] = -p[2]*u[1]*(1+u[1]) end nothing end # fix seeds seed = 100 Random.seed!(seed) numtraj= 100 seeds = rand(UInt, numtraj) W = WienerProcess(0.0,0.0,0.0) u0 = [-0.1f0,0.0f0] tspan = (0.0f0,100.0f0) p = [1.9f0,0.1f0] prob = SDEProblem(brusselator_f!,scalar_noise!,u0,tspan,p,noise=W) ensembleprob = EnsembleProblem(prob, prob_func = prob_func) sol = @time solve(ensembleprob,DRI1(),dt=0.1,EnsembleCPUArray(),trajectories=numtraj) sol_na = @time solve(ensembleprob,DRI1(),dt=0.8,adaptive=false,EnsembleCPUArray(),trajectories=numtraj) summ = EnsembleSummary(sol,0.0f0:0.5f0:100f0) pl = plot(summ,fillalpha=0.5,xlabel = \u0026quot;time t\u0026quot;, yaxis=\u0026quot;X(t)\u0026quot;, label= [\u0026quot;x₁(t)\u0026quot; \u0026quot;x₂(t)\u0026quot;], legend=true)  The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.\n  We can confirm Rößler\u0026rsquo;s observation4 that the adaptive scheme describes the time evolution of the SDE more accurately, as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.\nusing DifferentialEquations.EnsembleAnalysis meansol = timeseries_steps_mean(sol) meansol_na = timeseries_point_mean(sol_na,meansol.t) dts = [] tmp1 = tspan[1] for tmp2 in meansol.t global tmp1 push!(dts,tmp2-tmp1) tmp1 = tmp2 end # list_plots = [] for i in 1:length(meansol.u) l = @layout [a b] plt1 = plot(meansol[1, 1:i],meansol[2, 1:i], ylim = (-0.18, 0.18), xlim = (-0.13, 0.13), xlabel = \u0026quot;x₁(t)\u0026quot;, yaxis= \u0026quot;x₂(t)\u0026quot;, label=\u0026quot;adaptive\u0026quot;, lw=2, linecolor=1 ) plot!(meansol_na[1, 1:i],meansol_na[2, 1:i], ylim = (-0.18, 0.18), xlim = (-0.13, 0.13), xlabel = \u0026quot;x₁(t)\u0026quot;, yaxis= \u0026quot;x₂(t)\u0026quot;, label=\u0026quot;fixed step size\u0026quot;, lw=2, linecolor=2 ) pl2 = scatter(dts[1:i], xlabel = \u0026quot;step\u0026quot;, yaxis= \u0026quot;dtᵢ\u0026quot;, xlim = (0, length(meansol.u)), ylim = (0.0, 2.3), legend=false) plt = plot(plt1, pl2, layout = l) push!(list_plots, plt) end anim = animate(list_plots,lw=2,every=1)    GPU usage All necessary tools to accelerate the simulation of (stochastic) differential equations on GPUs within the SciML ecosystem are collected in the DiffEqGPU package.\nCurrently, bounds checking and return values are not allowed, i.e., functions must be written in the form:\nfunction f!(du,u,p,t) @inbounds begin du[1] = .. end nothing end  Except from those limitations, a user can specifiy ensemblealg=EnsembleGPUArray() to parallelize SDE solves across the GPU, see, e.g., the GPU tests for StochasticDiffEq for some examples. Note that for some high weak order solvers GPU usage is not recommended as scalar indexing is used.\nIf you have any questions or comments, please don’t hesitate to contact me!\n  Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. 23, Springer Science \u0026amp; Business Media (2013). \u0026#x21a9;\u0026#xfe0e;\n Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications 10 431-441 (1992). \u0026#x21a9;\u0026#xfe0e;\n Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics 59, 582–594 (2009). \u0026#x21a9;\u0026#xfe0e;\n Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation 77, 408-420 (2008) \u0026#x21a9;\u0026#xfe0e;\n   ","date":1597668395,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1597777200,"objectID":"43477ca9b2445f4eddd037d09515db97","permalink":"https://frankschae.github.io/post/high-weak/","publishdate":"2020-08-17T14:46:35+02:00","relpermalink":"/post/high-weak/","section":"post","summary":"This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.\nAfter an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.","tags":["GSoC 2020","julia","High weak order solver","SRK methods"],"title":"High weak order SDE solvers","type":"post"},{"authors":[],"categories":[],"content":"First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both new high weak order solvers as well as adjoint sensitivity methods for stochastic differential equations (SDEs). The project is embedded within the SciML organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software. Ultimately, the planned contributions will allow researchers to simulate (or even control) stochastic dynamics. Also inverse problems, where SDE models are fit to data, fall into the scope. Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.\nThis post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.\nHigh Weak Order Solvers Currently, the StochasticDiffEq package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense. In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.\nHowever in many situations, we are only aiming for computing an estimation for the expected value of the solution. In such situations, methods for the weak approximation are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods. High weak order solvers are particularly appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers having a smaller weak order convergence. As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.\nSecond order Runge-Kutta methods for Ito SDEs In the beginning of the community bonding period I finished the implementations of the DRI1()1 and RI1()2 methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes m. Specifically, only 2m-1 random variables have to be drawn (in contrast to m(m+1)/2 from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of m.\nAs an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise1:\n$$ \\scriptstyle d \\begin{pmatrix} X_1 \\\\ X_2 \\end{pmatrix} = \\begin{pmatrix} -\\frac{273}{512} \u0026amp; \\phantom{X_2}0 \\\\ -\\frac{1}{160} \\phantom{X_2} \u0026amp; -\\frac{785}{512}+\\frac{\\sqrt{2}}{8} \\end{pmatrix} \\begin{pmatrix} X_1 \\\\ X_2 \\end{pmatrix} dt + \\begin{pmatrix} \\frac{1}{4} X_1 \u0026amp; \\frac{1}{16} X_1 \\\\ \\frac{1-2\\sqrt{2}}{4} X_2 \u0026amp; \\frac{1}{10}X_1 +\\frac{1}{16} X_2 \\end{pmatrix} d \\begin{pmatrix} W_1 \\\\ W_2 \\end{pmatrix} $$\nwith initial value $$ X(t=0)= \\begin{pmatrix} 1 \\\\ 1\\end{pmatrix}.$$\nFor the function $f(x)=(x_1)^2$, we can analytically compute the expected value of the solution\n$$ \\rm{E}\\left[ f(X(t)) \\right] = \\exp(-t),$$\nwhich we use to test the weak convergence order of the algorithms in the following.\nTo compute the expected value numerically, we sample an ensemble of numtraj = 1e7 trajectories for different step sizes dt. The code for a single dt reads:\nusing StochasticDiffEq numtraj = 1e7 u₀ = [1.0,1.0] function f!(du,u,p,t) du[1] = -273//512*u[1] du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2] end function g!(du,u,p,t) du[1,1] = 1//4*u[1] du[1,2] = 1//16*u[1] du[2,1] = (1-2*sqrt(2))/4*u[1] du[2,2] = 1//10*u[1]+1//16*u[2] end dt = 1//8 tspan = (0.0,10.0) prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2)) h(z) = z^2 ensemble_prob = EnsembleProblem(prob; output_func = (sol,i) -\u0026gt; (h(sol[end][1]),false) ) sol = solve(ensemble_prob, DRI1(); dt=dt, save_start=false, save_everystep=false, weak_timeseries_errors=false, weak_dense_errors=false, trajectories=numtraj)  We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:\nLinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))  Repeating this procedure for some more values of dt, the log-log plot of the error as a function of dt displays nicely the second order convergence (slope $\\approx 2.2$).\n  In the next couple of weeks, my focus will be on\n adding other high weak order solvers, implementing adaptive time stepping.  More of our near-term goals are collected in this issue.\nAdjoint Sensitivity Methods for SDEs In parameter estimation/inverse problems, one is interested to know the optimal choice of parameters p such that a model f(p), e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error\n$$ L = |f(p)-y|^2. $$\nAn optimizer is then used to update the parameters $p$ such that $L$ is minimized. For this fit, local optimizers use the gradient $\\frac{dL}{dp}$ to minimize the loss function and ultimately solve the inverse problem. One possibility to obtain the gradient information for (stochastic) differential equations is to use automatic differentiation (AD). While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.\nAlternatively to the \u0026ldquo;direct\u0026rdquo; AD approaches, the adjoint sensitivity method can be used3. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs4. Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).\nIt turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an augmented state backwards in time launched at the end state of the forward evolution. In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. 4 are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the DiffEqSensitivity package for ODE adjoints earlier.\nReverse SDE time evolution The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms. As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed. In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values. After some fixes for the available noise processes, we are now able to reverse a stochastic time evolution either by using NoiseGrid which linearly interpolates between values of the noise on a given grid or by using a very general NoiseWrapper which interpolates in a distributionally-exact manner based on Brownian bridges.\nAs an example, the code below computes first the forward evolution of an SDE\n$$ dX = \\alpha X dt + \\beta X dW$$\nwith $\\alpha=1.01$, $\\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.\nusing StochasticDiffEq, DiffEqNoiseProcess α=1.01 β=0.87 dt = 1e-3 tspan = (0.0,1.0) u₀=1/2 tarray = collect(tspan[1]:dt:tspan[2]) f!(du,u,p,t) = du .= α*u g!(du,u,p,t) = du .= β*u prob = SDEProblem(f!,g!,[u₀],tspan) sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false) _sol = deepcopy(sol) # to make sure the plot is correct W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W)) prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1) sol1 = solve(prob1,EulerHeun(),dt=dt)    Gradients of diagonal SDEs I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.\nLet us consider again the linear SDE with multiplicative noise from above (with the same parameters). This SDE represents one of the few exact solvable cases. In the Stratonovich sense, the solution is given as\n$$ X(t) = X(0) \\exp(\\alpha t + \\beta W(t)).$$\nWe might be interested in optimizing the parameters $\\alpha$ and $\\beta$ to minimize a certain loss function acting on the solution $X(t)$. For such an optimization task, a useful search direction is indicated by the gradient of the loss function with respect to the parameters. The latter however requires the differentiation through the SDE solver \u0026ndash; if no analytical solution of the SDE is available.\nAs an example, let us consider a mean squared error loss\n$$ L(X(t)) = \\sum_i |X(t_i)|^2, $$\nacting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read\n$$ \\begin{aligned} \\frac{d L}{d \\alpha} \u0026amp;= 2 \\sum_i t_i |X(t_i)|^2 \\\\\n\\frac{d L}{d \\beta} \u0026amp;= 2 \\sum_i W(t_i) |X(t_i)|^2 \\end{aligned} $$\nfor $\\alpha$ and $\\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method\nusing Test, LinearAlgebra using DiffEqSensitivity, StochasticDiffEq using Random seed = 100 Random.seed!(seed) u₀ = [0.5] tstart = 0.0 tend = 0.1 dt = 0.005 trange = (tstart, tend) t = tstart:dt:tend tarray = collect(t) function g(u,p,t) sum(u.^2.0) end function dg!(out,u,p,t,i) (out.=-2.0*u) end p2 = [1.01,0.87] f(u,p,t) = p[1]*u σ(u,p,t) = p[2]*u Random.seed!(seed) prob = SDEProblem(f,σ,u₀,trange,p2) sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true) res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint()) noise = vec((@. sol.W(tarray))) Wextracted = [W[1][1] for W in noise] resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted)) resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted)) resp = [resp1, resp2] @test isapprox(res_p', resp, rtol = 1e-6) # True  With respect to the adjoint sensitivity methods, we are planning to\n finish the current backsolve adjoint version, allow for computing the gradients of non-commuting SDEs, implement also an interpolation adjoint version, benchmark it with respect to AD approaches  in the upcoming weeks. For more information, the interested reader might take a look at the open issues in the DiffEqSensitivity package.\nIf you have any questions or comments, please don’t hesitate to contact me!\n  Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics 59, 582–594 (2009). \u0026#x21a9;\u0026#xfe0e;\n Andreas Rößler, Journal on Numerical Analysis 47, 1713–1738 (2009). \u0026#x21a9;\u0026#xfe0e;\n Steven G. Johnson, \u0026ldquo;Notes on Adjoint Methods for 18.335.\u0026rdquo; Introduction to Numerical Methods (2012). \u0026#x21a9;\u0026#xfe0e;\n Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). \u0026#x21a9;\u0026#xfe0e;\n   ","date":1590844233,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1590844233,"objectID":"b18c7f652abe8e3ab1ef5f8324b079fc","permalink":"https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/","publishdate":"2020-05-30T15:10:33+02:00","relpermalink":"/post/gsoc2020-high-weak-order-solvers-sde-adjoints/","section":"post","summary":"First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.","tags":["GSoC 2020","julia","High weak order solver","SRK methods","Adjoint sensitivity methods"],"title":"GSoC 2020: High weak order SDE solvers and their utility in neural SDEs","type":"post"},{"authors":[],"categories":[],"content":"","date":1589109744,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1589109744,"objectID":"79e44c0f8a0e36807f6c8ada19854586","permalink":"https://frankschae.github.io/publication/njp/","publishdate":"2020-05-10T13:22:24+02:00","relpermalink":"/publication/njp/","section":"publication","summary":"Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.","tags":[],"title":"E. Greplova, A. Valenti, G. Boschung, F. Schäfer, N. Lörch, and S. Huber, New J. Phys. 22, 045003 (2020)","type":"publication"},{"authors":[],"categories":[],"content":"","date":1589109737,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1589109737,"objectID":"ce947d9c8d777c18560c094394319a35","permalink":"https://frankschae.github.io/publication/entropy/","publishdate":"2020-05-10T13:22:17+02:00","relpermalink":"/publication/entropy/","section":"publication","summary":"We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior of the particles launched either at the single-particle ground state or saddle-point energy, in a time-independent potential. We complement these results by a characterization of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time evolution of a single particle’s von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X)—which also allows us to extrapolate our results for increasing particle numbers.","tags":["Bosonic systems","Exact Diagonalization","MCTDH-X"],"title":"F. Schäfer, M. A. Bastarrachea-Magnani, A. U. J. Lode, L. de Forges de Parny, and A. Buchleitner, Entropy 22, 382 (2020)","type":"publication"},{"authors":[],"categories":[],"content":"","date":1589109731,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1589109731,"objectID":"919eb72cd3c1e9e9150d95b768cf58a8","permalink":"https://frankschae.github.io/publication/jphysb/","publishdate":"2020-05-10T13:22:11+02:00","relpermalink":"/publication/jphysb/","section":"publication","summary":"We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering cross section for an incoming plane wave, on the one hand, and to minimize the decay rate associated to a long-lived scattering resonance, on the other. In both cases, the optimum is achieved by configurations where all scatterers are placed on a line parallel to the direction of the incoming plane wave. The associated maximal scattering cross section increases quadratically with the number of scatterers for large N, whereas the minimal decay rate—which is realized by configurations that are not the same as those that maximize the scattering cross section—decreases exponentially as a function of N. Finally, we also analyze the stability of our optimized configurations with respect to small random displacements of the scatterers. These results demonstrate that optimized configurations of scatterers bear a considerable potential for applications such as quantum memories or mirrors consisting of only a few atoms.","tags":["multiple scattering","numerical optimization"],"title":"F. Schäfer, F. Eckert and T. Wellens, J. Phys. B 50, 235502 (2017)","type":"publication"},{"authors":[],"categories":[],"content":"","date":1589109296,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1589109296,"objectID":"d90fc2504c29d3f2758c889682fc049f","permalink":"https://frankschae.github.io/publication/pre99/","publishdate":"2020-05-10T13:14:56+02:00","relpermalink":"/publication/pre99/","section":"publication","summary":"We introduce an alternative method to identify phase boundaries in physical systems. It is based on training a predictive model such as a neural network to infer a physical system's parameters from its state. The deviation of the inferred parameters from the underlying correct parameters will be most susceptible and diverge maximally in the vicinity of phase boundaries. Therefore, peaks in the vector field divergence of the model's predictions are used as indication of phase transitions. Our method is applicable for phase diagrams of arbitrary parameter dimension and without prior information about the phases. Application to both the two-dimensional Ising model and the dissipative Kuramoto-Hopf model show promising results.","tags":["Machine Learning","Phase Transitions"],"title":"F. Schäfer and N. Lörch, Phys. Rev. E 99, 062107 (2019)","type":"publication"},{"authors":["FS in collaboration with Felix Eckert and Thomas Wellens"],"categories":[],"content":"We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering cross section for an incoming plane wave, on the one hand, and to minimize the decay rate associated to a long-lived scattering resonance, on the other. In both cases, the optimum is achieved by configurations where all scatterers are placed on a line parallel to the direction of the incoming plane wave. The associated maximal scattering cross section increases quadratically with the number of scatterers for large N, whereas the minimal decay rate—which is realized by configurations that are not the same as those that maximize the scattering cross section—decreases exponentially as a function of N. Finally, we also analyze the stability of our optimized configurations with respect to small random displacements of the scatterers. These results demonstrate that optimized configurations of scatterers bear a considerable potential for applications such as quantum memories or mirrors consisting of only a few atoms.\n","date":-62135596800,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":-62135596800,"objectID":"00a3646bc080207c242cecd1daa16abd","permalink":"https://frankschae.github.io/project/scattering/","publishdate":"0001-01-01T00:00:00Z","relpermalink":"/project/scattering/","section":"project","summary":"Numerical optimization of the positions of point scatterers to maximize the total scattering cross section for an incoming plane wave.","tags":["multiple scattering theory"],"title":"Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers","type":"project"}]
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+    [{"authors":null,"categories":null,"content":"I am a postdoc in the Julia Lab located in the Computer Science and Artificial Intelligence Laboratory (CSAIL) at the Massachusetts Institute of Technology (MIT). My research is focused on Scientific Machine Learning (SciML). I completed my PhD in physics in the Bruder group within the “Quantum Computing and Quantum Technology” PhD school at the University of Basel. During my PhD, I participated in the Google Summer of Code (GSoC) 2020 and 2021 programs with the projects “High weak order stochastic differential equation solvers and their utility in neural stochastic differential equations” within the Julia Language organization and “Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis” within the NumFocus organization, supervised by Chris Rackauckas, Moritz Schauer, Mohamed Tarek, and Yingbo Ma. Since 2020, I am a member of the SciML open source software organization for scientific machine learning.\n","date":1554595200,"expirydate":-62135596800,"kind":"term","lang":"en","lastmod":1554595200,"objectID":"2525497d367e79493fd32b198b28f040","permalink":"","publishdate":"0001-01-01T00:00:00Z","relpermalink":"","section":"authors","summary":"I am a postdoc in the Julia Lab located in the Computer Science and Artificial Intelligence Laboratory (CSAIL) at the Massachusetts Institute of Technology (MIT). My research is focused on Scientific Machine Learning (SciML).","tags":null,"title":"Frank Schäfer","type":"authors"},{"authors":[],"categories":null,"content":" Click on the Slides button above to view the built-in slides feature. Slides can be added in a few ways:\nCreate slides using Hugo Blox Builder’s Slides feature and link using slides parameter in the front matter of the talk file Upload an existing slide deck to static/ and link using url_slides parameter in the front matter of the talk file Embed your slides (e.g. Google Slides) or presentation video on this page using shortcodes. Further event details, including page elements such as image galleries, can be added to the body of this page.\n","date":1906549200,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1906549200,"objectID":"a8edef490afe42206247b6ac05657af0","permalink":"https://frankschae.github.io/talk/example-talk/","publishdate":"2017-01-01T00:00:00Z","relpermalink":"/talk/example-talk/","section":"event","summary":"An example talk using Hugo Blox Builder's Markdown slides feature.","tags":[],"title":"Example Talk","type":"event"},{"authors":["FS in collaboration with Gaurav Arya","Moritz Schauer","Ruben Seyer","Alex Lew","Mathieu Huot","Kartik Chandra","Vikash Mansinghka","Jonathan Ragan-Kelley","Chris Rackauckas"],"categories":[],"content":"Automatic differentiation (AD) has become ubiquitous throughout scientific computing and deep learning. However, AD systems have been restricted to the subset of programs that have a continuous dependence on parameters. Programs that have discrete stochastic behaviors governed by distribution parameters, such as flipping a coin with probability p of being heads, pose a challenge to these systems. In this work we develop a new AD methodology for programs with discrete randomness. We demonstrate how this method gives an unbiased and low-variance estimator.\n","date":1705333764,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1705333764,"objectID":"fd4013b59c325e7176a3ea5f9c2d78c3","permalink":"https://frankschae.github.io/project/stochasticad/","publishdate":"2024-01-15T10:49:24-05:00","relpermalink":"/project/stochasticad/","section":"project","summary":"We develop and implement AD algorithms for handling programs that can contain discrete randomness.","tags":[],"title":"Automatic Differentiation of Programs with Discrete Randomness","type":"project"},{"authors":["FS in collaboration with Flemming Holtorf","Julian Arnold","Chris Rackauckas","and Alan Edelman"],"categories":[],"content":"Control of devices at the quantum level holds enormous potential for current and future applications in the field of quantum information science. However, due to the nonlinear and stochastic nature of quantum systems under continuous observation, analytical solutions to all but the simplest quantum control problems remain unknown. In this project, we present a convex optimization framework to compute informative bounds on the best attainable control performance. Since our approach provides an under-approximator for the value function, we can use it directly to construct near-optimal heuristic controllers as demonstrated for a qubit subjected to homodyne detection and photon counting.\n","date":1682803609,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1682803609,"objectID":"d1e077164ce2a89141c07c546777d1ea","permalink":"https://frankschae.github.io/project/controlbounds/","publishdate":"2023-04-29T17:26:49-04:00","relpermalink":"/project/controlbounds/","section":"project","summary":"We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.","tags":["quantum control","convex optimization","stochastic optimal control","sum-of-squares"],"title":"Performance Bounds for Quantum Control","type":"project"},{"authors":[],"categories":[],"content":"","date":1673824837,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1673824837,"objectID":"91750f0e293b4386e71240dea8d36441","permalink":"https://frankschae.github.io/software/stochasticad/","publishdate":"2023-01-16T00:20:37+01:00","relpermalink":"/software/stochasticad/","section":"software","summary":"StochasticAD is an experimental, research package for automatic differentiation (AD) of stochastic programs. It implements AD algorithms for handling programs that can contain discrete randomness. Joint work with: [Gaurav Arya](https://github.com/gaurav-arya), [Moritz Schauer](https://github.com/mschauer), and [Chris Rackauckas](https://github.com/ChrisRackauckas).","tags":[],"title":"StochasticAD.jl: Automatic Differentiation of Programs with Discrete Randomness","type":"software"},{"authors":[],"categories":[],"content":"","date":1638487237,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1638487237,"objectID":"dd74b30c2376d17e758c3568ec2a62ad","permalink":"https://frankschae.github.io/software/abstractdifferentiation/","publishdate":"2021-12-03T00:20:37+01:00","relpermalink":"/software/abstractdifferentiation/","section":"software","summary":"Automatized generation of an extensive, unified, user-facing API for any AD package. Enables easy switching and composing between AD implementations. Joint work with [Mohamed Tarek](https://github.com/mohamed82008) and [other contributors](https://github.com/JuliaDiff/AbstractDifferentiation.jl/graphs/contributors).","tags":[],"title":"AbstractDifferentiation.jl: Backend-Agnostic Differentiable Programming in Julia","type":"software"},{"authors":[],"categories":[],"content":"Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.\nPossible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available on the GSoC page.\nBlog posts The following blog posts describe the work throughout the GSoC period in more detail:\nNeural Hybrid Differential Equations Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems Sensitivity Analysis of Hybrid Differential Equations AbstractDifferentiation.jl for AD-backend agnostic code Docs Documentation with respect to the adjoint sensitivity tools will be available on the local sensitivity analysis and on the control of automatic differentiation choices pages.\nAchievements Below is a list of PRs in the various repositories in chronological order.\nDiffEqSensitivity.jl Merged:\nAdd additive noise downstream test for DiffEqFlux DiscreteCallback fixes Allow for changes of p in callbacks Fix for using the correct uleft/pleft in continuous callback Fix broadcasting error on steady state adjoint Forward Least Squares Shadowing (LSS) Adjoint-mode for the LSS method concrete_solve dispatch for LSS methods Non-Intrusive Least Square Shadowing (NILSS) concrete_solve for NILSS Remove allocation in NILSS Handle additional callback case State-dependent Continuous Callbacks for BacksolveAdjoint QuadratureAdjoint() for ContinuousCallback More tests for Neural ODEs with callbacks for different sensitivity algorithms Support for PeriodicCallbacks in continuous adjoint methods AbstractDifferentiation.jl Merged:\nFixes gradient, Jacobian, Hessian, and vjp tests Open:\nAdd ForwardDiff and Zygote OrdinaryDiffEq.jl Merged:\nFix discrete reverse mode for some standard controllers DiffEqCallbacks.jl Merged:\nIntroduce a PeriodicCallbackAffect struct SteadyStateDiffEq.jl Merged:\nConvert alg.tspan to type of prob.u0 ChainRules.jl Merged:\nDo not differentiate through the construction of BitArray Use splatting in BitArray DiffEqNoiseProcess.jl Merged:\nAllow solvers to use Noise Grid with SVectors StochasticDiffEq.jl Merged:\nRemove Ihat2 matrix from weak solvers DiffEqDocs.jl Merged:\nSmall typo on plot page Add docs for shadowing methods Future work Besides the implementation of more shadowing methods, such as\nNILSAS, FD-NILSS, or Fast linear response, we are planning to\nbenchmark the new adjoints, refine the AbstractDifferentiation.jl package and use it within DiffEqSensitivity.jl, add more docs and examples. If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the Julia language discourse.\nAcknowledgement Many thanks to my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times. I would also like to thank Christoph Bruder, Julian Arnold, and Martin Koppenhöfer for helpful comments on my blog posts. Special thanks to Michael Poli and Stefano Massaroli for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google’s open source program for funding this experience!\n","date":1628883705,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1628883705,"objectID":"295f158a34d7a2a8bb042950210377c3","permalink":"https://frankschae.github.io/post/gsoc-2021/","publishdate":"2021-08-13T21:41:45+02:00","relpermalink":"/post/gsoc-2021/","section":"post","summary":"Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.","tags":["GSoC 2021"],"title":"Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis","type":"post"},{"authors":[],"categories":[],"content":"Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris’ article3 for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance and accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified4, for example, by focusing on learning only the parts of the model that are actually missing4 5. In the context of quantum control, we have demonstrated the power of this framework for closed6 and open quantum systems7.\n∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, a large variety of software implementations with different pros and cons exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by ChainRules.jl empowers this idea of a Glue AD where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.\nMohamed Tarek has started to implement a high level API for differentiation that unifies the APIs of all the AD packages in the Julia ecosystem. Ultimately, the API of our new package, AbstractDifferentiation.jl, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that DiffEqSensitivity.jl within the SciML software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I’ve started to fix remaining errors of the initial PR.\nThe interested reader is encouraged to look at Mohamed’s first PR for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.\nOptimization of the Rosenbrock function The Rosenbrock function is defined by\n$$ g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2. $$\nThe function $g$ has a global minimum at $(x_1^\\star, x_2^\\star)= (a, a^2)$ with $g(x_1^\\star, x_2^\\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.\nLet us now implement the Gauss–Newton algorithm to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\\bf{x}}=(x_1,\\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\\dots, f_M)$\n$$ S({\\bf x}) = \\frac{1}{2} \\sum_{i=1}^M f_i({\\bf x})^2. $$\nStarting from an initial guess ${\\bf x_0}$ for the minimum, the method runs through the iterations\n$$ {\\bf x}^{k+1} = {\\bf x}^k - \\alpha_k \\left(J^T J \\right)^{-1} J^T f({\\bf x}^k), $$ where $J$ is the Jacobian matrix at ${\\bf{x}}^k$ and $\\alpha_k$ is the step length determined via a line search subroutine.\nThe following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\\bf x^\\star}=(1,1)$ and our initial guess ${\\bf x_0}=(0,-0.1)$.\nusing Pkg path = @__DIR__ cd(path); Pkg.activate(\u0026#34;.\u0026#34;); Pkg.instantiate() ## AbstractDifferentiation is not released yet!! using AbstractDifferentiation using Test, LinearAlgebra using FiniteDifferences, ForwardDiff, Zygote using Enzyme, UnPack using Plots, LaTeXStrings # using Diffractor: ∂⃖¹ ## Diffractor needs \u0026gt;julia@1.6 ## Rosenbrock function # R: R^2 -\u0026gt; R: x -\u0026gt; (a-x₁)² + b(x₂-x₁²)² g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2 # visualization p = [1.0,100.0] x₀ = [0.0,-0.1] xopt = [1.0,1.0] do_plot = true if do_plot x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5 z = Surface((x₁,x₂)-\u0026gt;g([x₁,x₂],p), x₁, x₂) pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true, labelfontsize=20,camera = (3,50), xlabel = L\u0026#34;x_1\u0026#34;, ylabel = L\u0026#34;x_2\u0026#34;) pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp), labelfontsize=20, xlabel = L\u0026#34;x_1\u0026#34;, ylabel = L\u0026#34;x_2\u0026#34;) scatter!(pl2, [(x₀[1],x₀[2])], label=L\u0026#34;x_0\u0026#34;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green) scatter!(pl2, [(xopt[1],xopt[2])],label=L\u0026#34;x^\\star\u0026#34;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red) pl = plot(pl1,pl2, …","date":1627812197,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1627812197,"objectID":"d9ff4a599f5556fadd3c7cc978b481a3","permalink":"https://frankschae.github.io/post/abstract_differentiation/","publishdate":"2021-08-01T12:03:17+02:00","relpermalink":"/post/abstract_differentiation/","section":"post","summary":"Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris’ article3 for a detailed examination.","tags":["GSoC 2021","julia","Automatic Differentiation","AbstractDifferentiation.jl"],"title":"AbstractDifferentiation.jl for AD-backend agnostic code ","type":"post"},{"authors":["Frank Schäfer and Moritz Schauer"],"categories":[],"content":"In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations1. We highlight differences between explicit2 and implicit events3 4. As a paradigmatic example, we consider a bouncing ball described by the ODE\n$$ \\begin{aligned} \\text{d}z(t) \u0026amp;= v(t) \\text{d}t, \\\\ \\text{d}v(t) \u0026amp;= -\\mathrm g\\thinspace \\text{d}t \\end{aligned}\n$$\nwith initial condition\n$$ \\begin{aligned} z(t=0) \u0026amp;= z_0 = 5, \\\\ v(t=0) \u0026amp;= v_0 = -0.1. \\end{aligned}\n$$\nThe initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.\nSecond, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\\gamma=0.8$ (coefficient of restitution) that accounts for a imperfect elastic bounce on the ground.\nWhen ignoring the bounces, we can straightforwardly integrate the ODE analytically\n$$ \\begin{aligned} z(t) \u0026amp;= z_0 + v_0 t - \\frac{\\mathrm g}{2} t^2, \\\\ v(t) \u0026amp;= v_0 - \\mathrm g\\thinspace t \\end{aligned}\n$$\nor numerically using the OrdinaryDiffEq package from the SciML ecosystem.\n### simulate forward using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity using Plots, LaTeXStrings # dynamics function f(du,u,p,t) du[1] = u[2] du[2] = -p[1] end # parameters and solve z0 = 5.0 v0 = -0.1 t0 = 0.0 tend = 1.9 g = 10 γ = 0.8 u0 = [z0,v0] tspan = (t0,tend) p = [g, γ] prob = ODEProblem(f,u0,tspan,p) # plot forward trajectory sol = solve(prob,Tsit5(),saveat=0.1) pl = plot(sol, label = [\u0026#34;z(t)\u0026#34; \u0026#34;v(t)\u0026#34;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = \u0026#34;t\u0026#34;, legend=:bottomleft) hline!(pl, [0.0], label=false, color=\u0026#34;black\u0026#34;) savefig(pl,\u0026#34;BB_forward_no_bounce.png\u0026#34;) Of course, this way the ball continues to fall through the barrier at $z=0$.\nForward simulation with events At time $\\tau$ around $\\tau \\approx 1$, the ball hits the ground $z(\\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\\tau$ with new initial conditions\n$$ \\begin{aligned}\nz({\\tau}) \u0026amp;= z(\\tau-) ,\\\\ v({\\tau})\u0026amp;= -\\gamma v(\\tau-) , \\end{aligned}\n$$\nso that there is a jump in the velocity at the event time: the velocity right before the bounce, the left limit $v(\\tau-)$, and the velocity with which the ball continues its movement after the bounce $v(\\tau)$, are different.\nGiven our analytical solution for the state as a function of time, we can easily compute the event time $\\tau$ in terms of the initial condition and parameters as\n$$ \\tau = \\frac{v_0 + \\sqrt{v_0^2 + 2 \\mathrm g z_0}}{\\mathrm g}. $$\nExplicit events We can define the bounce of the ball as an explicit event by inserting the values of the initial condition and the parameters into the formula for $\\tau$. We obtain\n$$ \\tau = 0.99005. $$\nThe full explicit trajectory $z_{\\rm exp}(t) = z(t)$ is determined by\n$$ z(t) = \\begin{cases} z_0 + v_0 t - \\dfrac{\\mathrm g}{2} t^2 ,\u0026amp; \\forall t \u0026lt; \\tau, \\\\ -0.4901 \\mathrm g - 0.5 \\mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\\\ \\quad - (-0.99005 + t) (-0.99005 \\mathrm g + v_0)\\gamma ,\u0026amp; \\forall t \\ge \\tau, \\end{cases} $$\nwhere we used\n$$ \\begin{aligned}\nz({\\tau})\u0026amp;= z_0 + 0.99005 v_0 -0.4901 \\mathrm g, \\\\ v({\\tau})\u0026amp;= -\\gamma v({\\tau-}) = -\\gamma(v_0 - 0.99005 \\mathrm g) . \\end{aligned}\n$$\nHere the change in state $(z,v)$ at the event time is defined with the help of an affect function\n$$ a(z,v) = (z, -\\gamma v). $$\nNumerically, we use a DiscreteCallback in this case to simulate the system.\n# solve with DiscreteCallback (explicit event) tstar = (v0 + sqrt(v0^2+2*z0*g))/g condition1(u,t,integrator) = (t == tstar) affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2] cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true)) sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar]) Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.\nImplicit events The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)\n$$ g(z,v,p,t), $$\nwhere an event occurs at time $\\tau$ if $g(z(\\tau),v(\\tau),p,\\tau) = 0$. We have already used this condition to define our impact time $\\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time $g(z,v,p,t) = 0$.\nAs in the previous case, we can analytically compute the full trajectory of the ball. By …","date":1626434644,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1626434644,"objectID":"a60e1fb7808c483d902fff1d56e2a3a3","permalink":"https://frankschae.github.io/post/bouncing_ball/","publishdate":"2021-07-16T13:24:04+02:00","relpermalink":"/post/bouncing_ball/","section":"post","summary":"In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.","tags":["GSoC 2021","Hybrid differential equations","Adjoint sensitivity methods","Event handling"],"title":"Sensitivity Analysis of Hybrid Differential Equations","type":"post"},{"authors":[],"categories":[],"content":"In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the “butterfly effect”. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.\nSensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control1 2 3. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing4 (LSS) or non-intrusive least squares shadowing5 (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem – the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the DiffEqSensitivity.jl package.\nThe objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:\n$$ \\langle g \\rangle_∞ = \\lim_{T \\rightarrow ∞} \\langle g \\rangle_T, $$ where $$ \\langle g \\rangle_T = \\frac{1}{T} \\int_0^T g(u,s) \\text{d}t. $$ Under the assumption of ergodicity, $\\langle g \\rangle_∞$ only depends on $p$.\nThe Lorenz system One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\\rho$, $\\sigma$, and $\\beta$. Its time evolution is given by the ODE:\n$$ \\begin{pmatrix} \\text{d}x \\\\ \\text{d}y \\\\ \\text{d}z \\\\ \\end{pmatrix} = \\begin{pmatrix} \\sigma (y-x)\\\\ x(\\rho-z) - y\\\\ x y - \\beta z \\\\ \\end{pmatrix}\\text{d}t $$\nFor simplicity, let us fix $\\sigma=10$ and $\\beta=8/3$ and focus only on the sensitivity with respect to $\\rho$. The classic Lorenz attractor is obtained when using $\\rho=28$:\nusing Random; Random.seed!(1234) using OrdinaryDiffEq using Statistics using QuadGK, ForwardDiff, Calculus using DiffEqSensitivity using SparseArrays, LinearAlgebra # simulate 1 trajectory of the Lorenz system forward function lorenz!(du,u,p,t) du[1] = 10*(u[2]-u[1]) du[2] = u[1]*(p[1]-u[3]) - u[2] du[3] = u[1]*u[2] - (8//3)*u[3] end p = [28.0] tspan_init = (0.0,30.0) tspan_attractor = (30.0,50.0) u0 = rand(3) prob_init = ODEProblem(lorenz!,u0,tspan_init,p) sol_init = solve(prob_init,Tsit5()) prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p) sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14) using Plots, LaTeXStrings pl1 = plot(sol_init,vars=(1,2,3), legend=true, label = \u0026#34;initial\u0026#34;, labelfontsize=20, lw = 2, xlabel = L\u0026#34;x\u0026#34;, ylabel = L\u0026#34;y\u0026#34;, zlabel = L\u0026#34;z\u0026#34;, xlims=(-25,30),ylims=(-30,30),zlims=(5,49) ) plot!(pl1, sol_attractor,vars=(1,2,3), label=\u0026#34;attractor\u0026#34;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49) ) savefig(pl1, \u0026#34;Lorenz_forward.png\u0026#34;) Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.\nFollowing Refs.4 and 5, we choose\n$$ \\langle z \\rangle_∞ = \\lim_{T \\rightarrow ∞} \\frac{1}{T} \\int_0^T z \\text{d}t $$\nas the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\\rho$.\nfunction compute_objective(sol) quadgk(t-\u0026gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1] end pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L\u0026#34;z(t)\u0026#34;, xlabel=L\u0026#34;t\u0026#34;, label=false, labelfontsize=20,lw = 2) mean_z = [mean(getindex.(sol_attractor.u,3))] int_z = compute_objective(sol_attractor) hline!(pl2, [int_z], label=L\u0026#34;\\langle z\\rangle\u0026#34;, lw = 2) savefig(pl2, \u0026#34;zsingle.png\u0026#34;) # for each value of the parameter, solve 20 times the initial value problem # wrap the procedure inside a function depending on p function Lorenz_solve(p) u0 = rand(3) prob_init = ODEProblem(lorenz!,u0,tspan_init,p) sol_init = solve(prob_init,Tsit5()) prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p) sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14) sol_attractor, prob_attractor end Niter = 10 ps = collect(0.0:1.0:50.0) probs = [] sols = [] zmean = [] zstd = [] for ρ in ps @show ρ ztmp = [] for i=1:Niter sol, prob = Lorenz_solve([ρ]) zbar = compute_objective(sol) push!(sols, sol) push!(probs, prob) push!(ztmp, zbar) end …","date":1625216902,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1625216902,"objectID":"263091e46f2c8aead98407a523d8a2fe","permalink":"https://frankschae.github.io/post/shadowing/","publishdate":"2021-07-02T11:08:22+02:00","relpermalink":"/post/shadowing/","section":"post","summary":"In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.","tags":["GSoC 2021","julia","Adjoint sensitivity methods","Forward sensitivity methods","Shadowing","Chaotic systems"],"title":"Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems","type":"post"},{"authors":[],"categories":[],"content":"I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek. This year’s project is embedded within the NumFocus/SciML organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.\nThis first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).\nNote: Please check the SciMLSensitivity.jl docs for a maintained neural hybrid DE tutorial!\nHybrid Differential Equations DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by callbacks. Evidently, the incorporation of discontinuities allows a user to specify changes (events) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by DiscreteCallbacks, implicit events have to be specified by ContinuousCallbacks. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to 0. Thus, implicit events require some sort of rootfinding procedure.\nSome relevant examples for hybrid DEs with discrete or continuous callbacks are:\nquantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.1 (ContinuousCallback). a bouncing ball2 (ContinuousCallback). classical point process models, such as a Poisson process3. digital controllers4, where a continuous system dynamics is controlled by a discrete-time controller (DiscreteCallback). pharmacokinetic models5, where explicit dosing times change the drug concentration in the blood (DiscreteCallback). The simplest possible example being the one-compartment model. kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (DiscreteCallback). The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. 2 and 3.\nKicked harmonic oscillator Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form\n$$ \\ddot{x}(t) + a\\cdot\\dot{x}(t) + b \\cdot x(t) = 0 , $$\nwhere $a=0.1$ and $b=1$ with initial conditions\n$$ \\begin{aligned} x(t=0) \u0026amp;= 1 \\\\ v(t=0) \u0026amp;= \\dot{x}(t=0) = 0. \\end{aligned}\n$$\nThis second-order ODE can be reduced to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the DifferentialEquations.jl package. (Instead of doing this reduction manually, we could also use ModelingToolkit.jl to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a SecondOrderODEProblem implemented.) The Julia code reads:\nusing DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity using Zygote using Random u0 = Float32[1.; 0.] tspan = (0.0f0,50.0f0) dtsave = 0.5f0 t = tspan[1]:dtsave:tspan[2] function oscillator!(du,u,p,t) du[1] = u[2] du[2] = - u[1] - 1//10*u[2] return nothing end prob_data = ODEProblem(oscillator!,u0,tspan) # ODE without kicks pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[\u0026#34;x(t)\u0026#34; \u0026#34;v(t)\u0026#34;]) We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as 1.\nkicktimes = tspan[1]:1:tspan[2] function kick!(integrator) integrator.u[end] += one(eltype(integrator.u)) end cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false)) sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t) t_data = sol_data.t ode_data = Array(sol_data) # visualize data pl1 = plot(t_data,ode_data[1,:],label=\u0026#34;data x(t)\u0026#34;) plot!(pl1,t_data,ode_data[2,:],label=\u0026#34;data v(t)\u0026#34;) pl2 = plot(t_data[1:20],ode_data[1,1:20],label=\u0026#34;data x(t)\u0026#34;) plot!(pl2,t_data[1:20],ode_data[2,1:20],label=\u0026#34;data v(t)\u0026#34;) pl = plot(pl2, pl1, layout=(1,2), xlabel=\u0026#34;t\u0026#34;) The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting save_positions=(true,true), the kicks would be saved before and after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations6, we now aim at learning (potentially) missing parts of the model from these data traces.\nHigh domain knowledge For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the affect! function of the event only acts on the …","date":1623847817,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1623847817,"objectID":"c5c641840c6146609c2e90c018c85a3f","permalink":"https://frankschae.github.io/post/hybridde/","publishdate":"2021-06-16T14:50:17+02:00","relpermalink":"/post/hybridde/","section":"post","summary":"I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.","tags":["GSoC 2021","julia","Hybrid differential equations","Adjoint sensitivity methods","Event handling"],"title":"Neural Hybrid Differential Equations","type":"post"},{"authors":[],"categories":[],"content":"","date":1615332097,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615332097,"objectID":"6b8484ee245fb6a73711cfc61c673627","permalink":"https://frankschae.github.io/software/sciml/","publishdate":"2021-03-10T00:21:37+01:00","relpermalink":"/software/sciml/","section":"software","summary":"Contributions to the SciML ecosystem in Julia, especially the DiffEqSensitivity.jl package for sensitivity analysis utilities, the StochasticDiffEq.jl package for stochastic differential equations solvers, and the DiffEqNoiseProcess package for tools to develop noise processes for differential equations.","tags":[],"title":"SciML Scientific Machine Learning Software","type":"software"},{"authors":[],"categories":[],"content":"","date":1615332037,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615332037,"objectID":"7d136c46f74da06acca4c2ac00a7b219","permalink":"https://frankschae.github.io/software/mitosisstochasticdiffeq/","publishdate":"2021-03-10T00:20:37+01:00","relpermalink":"/software/mitosisstochasticdiffeq/","section":"software","summary":"Implementation of the backward filter and the forward change of measure of the Automatic Backward Filtering Forward Guiding paradigm. Joint work with [Moritz Schauer](https://github.com/mschauer).","tags":[],"title":"MitosisStochasticDiffEq.jl","type":"software"},{"authors":["FS in collaboration with Pavel Sekatski","Martin Koppenhöfer","Niels Lörch","Christoph Bruder","and Michal Kloc"],"categories":[],"content":"Conceptually, it is straightforward to determine the time evolution of a quantum system for a fixed initial state given its (time-dependent) Hamiltonian or Lindbladian. Depending on the physical context, the dynamics is described by an ordinary or stochastic differential equation. In quantum state control, which is of paramount importance for quantum computation, we aim at solving the inverse problem. That is, starting from a distribution of initial states, we seek protocols that allow us to reach a desired target state by optimization of free parameters of the differential equation (control drives) in a certain time interval. To solve this control problem, we implement the system dynamics as part of a fully differentiable program and use a loss function that quantifies the distance from the target state. Specifically, we employ a neural network that maps an observation of the state of the qubit to a control drive defined via the differential equation for each time interval. To implement efficient training, we backpropagate the gradient information from the loss function through the SDE solver using adjoint sensitivity methods. Such a procedure should ultimately combine powerful tools from machine learning and scientific computation.\n","date":1615332029,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615332029,"objectID":"0dda26cfe2920348a8f4a2cf6ef7ac1d","permalink":"https://frankschae.github.io/project/dp_for_control/","publishdate":"2021-03-10T00:20:29+01:00","relpermalink":"/project/dp_for_control/","section":"project","summary":"Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.","tags":["SciML","differentiable programming","NODE","NSDE","adjoint sensitivity methods","automatic differentiation","quantum control"],"title":"Control of (Stochastic) Quantum Dynamics with Differentiable Programming","type":"project"},{"authors":["FS in collaboration with Eliska Greplova","Tomoki Hirosawa","Jozef Bucko","František Herman","Rebekka Garreis","Chuyao Tong","Annika Kurzmann","Thomas Ihn","Hideaki Maebashi","Hiroyasu Matsuura","and Masao Ogata"],"categories":[],"content":"","date":1615330603,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615330603,"objectID":"94d4899d49aece16c0d03e7ab9c4963f","permalink":"https://frankschae.github.io/project/reconstruction/","publishdate":"2021-03-09T23:56:43+01:00","relpermalink":"/project/reconstruction/","section":"project","summary":"We address the challenges of connecting measurements of a physical system to an underlying theoretical description.","tags":[],"title":"Automated Data-driven Reconstruction of Physical Properties","type":"project"},{"authors":["FS in collaboration with Julian Arnold","Flemming Holtorf","Eliska Greplova","Agnes Valenti","Martin Zonda","Axel Lode","Gregor Boschung","Sebastian Huber","and Niels Lörch"],"categories":[],"content":"Artificial neural networks have successfully been used to identify phase transitions from data and classify data into distinct phases in an automated fashion. The power and success of these approaches (e.g., “learning by confusion” or the “prediction-based method”) can be attributed to the ability of deep neural networks to learn arbitrary functions. However, the larger a neural network, the more computational resources are needed to train it, and the more difficult it is to understand its decision making. In this project, we establish a solid theoretical foundation of popular machine-learning methods that rely on neural networks for detecting phase transitions and develop new efficient numerical routines to identify phase transitions directly from data. In the future, our methods could make it possible to detect as yet unknown phase transitions, for instance in quantum simulators or in novel materials.\n","date":1615330603,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615330603,"objectID":"96c9e3dc2a1485e968cc89f130ac8af0","permalink":"https://frankschae.github.io/project/ml_for_pt/","publishdate":"2021-03-09T23:56:43+01:00","relpermalink":"/project/ml_for_pt/","section":"project","summary":"Data-driven methods based on sample instances of the state of a physical system as a function of the system's parameters.","tags":["Machine learning","analytical predictors","neural networks","unsupervised learning"],"title":"Machine Learning for Phase Transitions","type":"project"},{"authors":["FS in collaboration with Laurent de Forges de Parny","Miguel Bastarrachea","Axel Lode","and Andreas Buchleitner"],"categories":[],"content":"We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior of the particles launched either at the single-particle ground state or saddle-point energy, in a time-independent potential. We complement these results by a characterization of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time evolution of a single particle’s von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X)—which also allows us to extrapolate our results for increasing particle numbers.\n","date":1615282751,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1615282751,"objectID":"4b16c3ed4d0b9e975df9b36f75807256","permalink":"https://frankschae.github.io/project/mbdynamics/","publishdate":"2021-03-09T10:39:11+01:00","relpermalink":"/project/mbdynamics/","section":"project","summary":"Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.","tags":["Many-body physics","exact diagonalization","Bose-Hubbard model","MCTDH-X"],"title":"Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well ","type":"project"},{"authors":[],"categories":[],"content":"Project summary In this project, we have implemented new promising tools within the SciML organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations. The high weak order solvers will allow for massive performance advantages for fitting expectations of equations. Instead of automatic differentiation (AD) through the operations of an SDE solver, which scales poorly in memory, one can now use efficient stochastic adjoint sensitivity methods.\nBlog posts The following posts describe the work during the entire period in more detail:\nGSoC 2020: High weak order SDE solvers and their utility in neural SDEs High weak order SDE solvers Docs The documentation of the solvers is available here. Docs with respect to the adjoint sensitivity tools will be available here.\nAchievements Please find below a list of the PRs carried out during GSoC in the different repositories in chronological order within the SciML ecosystem.\nStochasticDiffEq.jl Merged:\nInplace version of DRI1 scheme RI1 method tstop fixes for reverse time propagation Fixes for EulerHeun scheme Speed up the tests for the DRI1 and RI1 schemes RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes RDI1WM scheme Adaptive version of the stochastic Runge-Kutta schemes by embedding RS1 and RS2 schemes PL1WM scheme NON scheme Citations for weak methods Stochastic improved and modified Euler methods COM scheme Computationally more efficient NON variant (NON2) Open:\nStatic array tests Levy area for non-commutative noise processes DiffEqSensitivity.jl Merged:\nAdjoint sensitivities for steady states Concrete_solve dispatch for steady state problem BacksolveAdjoint for SDEs GPU savety for SDE BacksolveAdjoint Tests for concrete solve with respect to SDEs Alternative differentiation choices (vjps) for noise Jacobian Fixes and tests for inplace formulation of BacksolveAdjoint Efficient SDE BacksolveAdjoint for scalar noise Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms InterpolatingAdjoint for SDEs Citations for backsolve, steadystate and interpolation adjoint Allow for more general noise processes: replace NoiseGrid by NoiseWrapper Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs Cheaper non-diagonal noise tests Open:\nSupport adjoints for SDEs written in the Ito sense DiffEqNoiseProcess.jl Merged:\nMulti-dimensional Brownian motion tests Bug fix for inplace form of NoiseGrid Reversible NoiseWrapper Relax the size constraints of the available noise processes Generalization of the real-valued white noise process function Fix of an extraction issue with NoiseGrid Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory Extraction and endpoint fixes on NoiseWrapper Reversal of SDEs written in the Ito sense DiffEqGPU.jl Merged:\nMemory efficient reduction function for ensemble problems ModelingToolkit.jl Merged:\nmodelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense DiffEqDevTools.jl Merged:\nNoiseWrapper alternative for analyticless convergence tests of SDE solvers test_convergence() dispatch for ensemble simulations Work precision set for ensemble problems DiffEqBase.jl Merged:\nFix concrete_solve tests Future work There is still a lot that we’d like to do, e.g.,\nWriting up more docs and examples Implementing drift-implicit weak stochastic Runge-Kutta solvers Finishing the SDE adjoints for the Ito sense Implementing a virtual Brownian tree to store the noise processes in O(1) memory Setting up an OptimalControl library that allows for easy usage of the new tools within a symbolic interface Benchmarking of the new solvers and adjoints Contributions, suggestions \u0026amp; comments are always welcome! You might like to join our slac channels #diffeq-bridged and #neuralsde to get in touch.\nAcknowledgement I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback. I would also like to thank Christoph Bruder, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts. Many thanks to the very supportive julia community and to Google’s open source program for funding this experience!\n","date":1598475589,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1598475589,"objectID":"c74c1755693bb9cf28d7bf9019cfb1ee","permalink":"https://frankschae.github.io/post/gsoc-2020/","publishdate":"2020-08-26T22:59:49+02:00","relpermalink":"/post/gsoc-2020/","section":"post","summary":"Project summary In this project, we have implemented new promising tools within the SciML organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.","tags":["GSoC 2020"],"title":"High weak order solvers and adjoint sensitivity analysis for stochastic differential equations","type":"post"},{"authors":[],"categories":[],"content":"This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.\nAfter an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process. Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics. Finally, we discuss how to run simulations on GPU hardware.\nThroughout this post, we shall use the vector notation $X(t)$ to denote the solution of the d-dimensional Ito SDE system\n$$ dX(t) = a(t,X(t)) dt + b(t,X(t)) dW $$\nwith an m-dimensional driving Wiener process W(t) in the time span $\\mathbb{I}=[t_0, T]$, where $a: \\mathbb{I}\\times\\mathbb{R}^d \\rightarrow \\mathbb{R}^d$ and $b: \\mathbb{I}\\times \\mathbb{R}^{d} \\rightarrow \\mathbb{R}^{d \\times m}$ are continuous functions which fulfill a global Lipschitz condition.1 For simplicity, we write $X(t)$ for both time-discrete approximations and continuous-time random variables in the following.\nStrong convergence Suppose that we encounter the following problem: Given noisy observations $Z(t)$ (e.g., originating from measurement noise), what is the best estimate $\\hat{X}(t)$ of a stochastic system $X(t)$ satisfying the form above. Intuitively, we aim at filtering away the noise from the observations in an optimal way. Such tasks are well known as filtering problems.\nTo solve a filtering problem, we need a solver whose sample paths $Y(t)$ are close to the ones of the stochastic process $X(t)$, i.e., the solver should allow us to reconstruct correctly the numerical solution of each single trajectory of an SDE.\nIntroducing the absolute error at the final time $T$ as $$ \\rm{E}(|X(T) -Y(T)|) \\leq \\sqrt{\\rm{E}(|X(T)-Y(T)|^2)}, $$ we define convergence in the strong sense with order $p$ of a time discrete approximation $Y(T)$ with step size $h$ to the solution $X(T)$ of a SDE at time $T$ if there exists a constant $C$ (independent of $h$) and a constant $\\delta \u0026gt; 0$ such that $$ \\rm{E}(|X(T) -Y(T)|) \\leq C \\cdot h^p, $$ for each $h \\in [0, \\delta]$.\nThe StochasticDiffEq package contains various state-of-the-art solvers for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms. For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI) can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where b has only entries on its diagonal or $m=1$. The main difficulty for the construction of strong methods with an order \u0026gt; 1/2 arises from the need of an accurate estimation of multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of dW in the case of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed in the case of non-commutative noise processes.2 Currently, we are also implementing those iterated integrals in the StochasticDiffEq library.\nWeak convergence Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the expected value of the solution in many situations. In these cases, methods for the weak approximation are sufficient and – due to the less restrictive formulation of the objective – those solvers are computationally cheaper than their strong counterparts. For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics. Our new contributions are particularly appealing for many-body simulations, which are the computationally most demanding problems in quantum mechanics.\nWe define convergence in the weak sense with order $p$ of a time-discrete approximation $Y(T)$ with step size $h$ to the solution $X(T)$ of a SDE at time $T$ if there exists a constant $C$ (independent of $h$) and a constant $\\delta \u0026gt; 0$ such that $$ |\\rm{E}(g(X(T))) -\\rm{E}(g(Y(T)))| \\leq C \\cdot h^p, $$ $~$\nfor any polynomial $g$ for each $h \\in [0, \\delta]$.\nWe demonstrate below that high weak order solvers are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.\nNew high weak order methods A list of all new weak solvers is available in the SciML documentation. Note that we also implemented methods designed for the Stratonovich sense. For the subsequent examples regarding Ito SDEs, we use only a subset …","date":1597668395,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1597777200,"objectID":"43477ca9b2445f4eddd037d09515db97","permalink":"https://frankschae.github.io/post/high-weak/","publishdate":"2020-08-17T14:46:35+02:00","relpermalink":"/post/high-weak/","section":"post","summary":"This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.\nAfter an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.","tags":["GSoC 2020","julia","High weak order solver","SRK methods"],"title":"High weak order SDE solvers","type":"post"},{"authors":[],"categories":[],"content":"First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both new high weak order solvers as well as adjoint sensitivity methods for stochastic differential equations (SDEs). The project is embedded within the SciML organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software. Ultimately, the planned contributions will allow researchers to simulate (or even control) stochastic dynamics. Also inverse problems, where SDE models are fit to data, fall into the scope. Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.\nThis post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.\nHigh Weak Order Solvers Currently, the StochasticDiffEq package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense. In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.\nHowever in many situations, we are only aiming for computing an estimation for the expected value of the solution. In such situations, methods for the weak approximation are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods. High weak order solvers are particularly appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers having a smaller weak order convergence. As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.\nSecond order Runge-Kutta methods for Ito SDEs In the beginning of the community bonding period I finished the implementations of the DRI1()1 and RI1()2 methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes m. Specifically, only 2m-1 random variables have to be drawn (in contrast to m(m+1)/2 from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of m.\nAs an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise1:\n$$ \\scriptstyle d \\begin{pmatrix} X_1 \\\\ X_2 \\end{pmatrix} = \\begin{pmatrix} -\\frac{273}{512} \u0026amp; \\phantom{X_2}0 \\\\ -\\frac{1}{160} \\phantom{X_2} \u0026amp; -\\frac{785}{512}+\\frac{\\sqrt{2}}{8} \\end{pmatrix} \\begin{pmatrix} X_1 \\\\ X_2 \\end{pmatrix} dt + \\begin{pmatrix} \\frac{1}{4} X_1 \u0026amp; \\frac{1}{16} X_1 \\\\ \\frac{1-2\\sqrt{2}}{4} X_2 \u0026amp; \\frac{1}{10}X_1 +\\frac{1}{16} X_2 \\end{pmatrix} d \\begin{pmatrix} W_1 \\\\ W_2 \\end{pmatrix} $$\nwith initial value $$ X(t=0)= \\begin{pmatrix} 1 \\\\ 1\\end{pmatrix}.$$\nFor the function $f(x)=(x_1)^2$, we can analytically compute the expected value of the solution\n$$ \\rm{E}\\left[ f(X(t)) \\right] = \\exp(-t),$$\nwhich we use to test the weak convergence order of the algorithms in the following.\nTo compute the expected value numerically, we sample an ensemble of numtraj = 1e7 trajectories for different step sizes dt. The code for a single dt reads:\nusing StochasticDiffEq numtraj = 1e7 u₀ = [1.0,1.0] function f!(du,u,p,t) du[1] = -273//512*u[1] du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2] end function g!(du,u,p,t) du[1,1] = 1//4*u[1] du[1,2] = 1//16*u[1] du[2,1] = (1-2*sqrt(2))/4*u[1] du[2,2] = 1//10*u[1]+1//16*u[2] end dt = 1//8 tspan = (0.0,10.0) prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2)) h(z) = z^2 ensemble_prob = EnsembleProblem(prob; output_func = (sol,i) -\u0026gt; (h(sol[end][1]),false) ) sol = solve(ensemble_prob, DRI1(); dt=dt, save_start=false, save_everystep=false, weak_timeseries_errors=false, weak_dense_errors=false, trajectories=numtraj) We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:\nLinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2])) Repeating this procedure for some more values of dt, the log-log plot of the error as a function of dt displays nicely the second order convergence (slope $\\approx 2.2$).\nIn the next couple of weeks, my focus will be on\nadding …","date":1590844233,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1590844233,"objectID":"b18c7f652abe8e3ab1ef5f8324b079fc","permalink":"https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/","publishdate":"2020-05-30T15:10:33+02:00","relpermalink":"/post/gsoc2020-high-weak-order-solvers-sde-adjoints/","section":"post","summary":"First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.","tags":["GSoC 2020","julia","High weak order solver","SRK methods","Adjoint sensitivity methods"],"title":"GSoC 2020: High weak order SDE solvers and their utility in neural SDEs","type":"post"},{"authors":[],"categories":[],"content":"","date":1589109744,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1589109744,"objectID":"79e44c0f8a0e36807f6c8ada19854586","permalink":"https://frankschae.github.io/publication/njp/","publishdate":"2020-05-10T13:22:24+02:00","relpermalink":"/publication/njp/","section":"publication","summary":"Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.","tags":[],"title":"E. Greplova, A. Valenti, G. Boschung, F. Schäfer, N. Lörch, and S. Huber, New J. Phys. 22, 045003 (2020)","type":"publication"},{"authors":[],"categories":[],"content":"","date":1589109737,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1589109737,"objectID":"ce947d9c8d777c18560c094394319a35","permalink":"https://frankschae.github.io/publication/entropy/","publishdate":"2020-05-10T13:22:17+02:00","relpermalink":"/publication/entropy/","section":"publication","summary":"We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior of the particles launched either at the single-particle ground state or saddle-point energy, in a time-independent potential. We complement these results by a characterization of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time evolution of a single particle’s von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X)—which also allows us to extrapolate our results for increasing particle numbers.","tags":["Bosonic systems","Exact Diagonalization","MCTDH-X"],"title":"F. Schäfer, M. A. Bastarrachea-Magnani, A. U. J. Lode, L. de Forges de Parny, and A. Buchleitner, Entropy 22, 382 (2020)","type":"publication"},{"authors":[],"categories":[],"content":"","date":1589109731,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1589109731,"objectID":"919eb72cd3c1e9e9150d95b768cf58a8","permalink":"https://frankschae.github.io/publication/jphysb/","publishdate":"2020-05-10T13:22:11+02:00","relpermalink":"/publication/jphysb/","section":"publication","summary":"We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering cross section for an incoming plane wave, on the one hand, and to minimize the decay rate associated to a long-lived scattering resonance, on the other. In both cases, the optimum is achieved by configurations where all scatterers are placed on a line parallel to the direction of the incoming plane wave. The associated maximal scattering cross section increases quadratically with the number of scatterers for large N, whereas the minimal decay rate—which is realized by configurations that are not the same as those that maximize the scattering cross section—decreases exponentially as a function of N. Finally, we also analyze the stability of our optimized configurations with respect to small random displacements of the scatterers. These results demonstrate that optimized configurations of scatterers bear a considerable potential for applications such as quantum memories or mirrors consisting of only a few atoms.","tags":["multiple scattering","numerical optimization"],"title":"F. Schäfer, F. Eckert and T. Wellens, J. Phys. B 50, 235502 (2017)","type":"publication"},{"authors":[],"categories":[],"content":"","date":1589109296,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1589109296,"objectID":"d90fc2504c29d3f2758c889682fc049f","permalink":"https://frankschae.github.io/publication/pre99/","publishdate":"2020-05-10T13:14:56+02:00","relpermalink":"/publication/pre99/","section":"publication","summary":"We introduce an alternative method to identify phase boundaries in physical systems. It is based on training a predictive model such as a neural network to infer a physical system's parameters from its state. The deviation of the inferred parameters from the underlying correct parameters will be most susceptible and diverge maximally in the vicinity of phase boundaries. Therefore, peaks in the vector field divergence of the model's predictions are used as indication of phase transitions. Our method is applicable for phase diagrams of arbitrary parameter dimension and without prior information about the phases. Application to both the two-dimensional Ising model and the dissipative Kuramoto-Hopf model show promising results.","tags":["Machine Learning","Phase Transitions"],"title":"F. Schäfer and N. Lörch, Phys. Rev. E 99, 062107 (2019)","type":"publication"},{"authors":["Frank Schäfer"],"categories":null,"content":" Create your slides in Markdown - click the Slides button to check out the example. Add the publication’s full text or supplementary notes here. You can use rich formatting such as including code, math, and images.\n","date":1554595200,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1554595200,"objectID":"557dc08fd4b672a0c08e0a8cf0c9ff7d","permalink":"https://frankschae.github.io/publication/preprint/","publishdate":"2017-01-01T00:00:00Z","relpermalink":"/publication/preprint/","section":"publication","summary":"Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum.","tags":["Source Themes"],"title":"An example preprint / working paper","type":"publication"},{"authors":[],"categories":[],"content":"Create slides in Markdown with Hugo Blox Builder Hugo Blox Builder | Documentation\nFeatures Efficiently write slides in Markdown 3-in-1: Create, Present, and Publish your slides Supports speaker notes Mobile friendly slides Controls Next: Right Arrow or Space Previous: Left Arrow Start: Home Finish: End Overview: Esc Speaker notes: S Fullscreen: F Zoom: Alt + Click PDF Export Code Highlighting Inline code: variable\nCode block:\nporridge = \u0026#34;blueberry\u0026#34; if porridge == \u0026#34;blueberry\u0026#34;: print(\u0026#34;Eating...\u0026#34;) Math In-line math: $x + y = z$\nBlock math:\n$$ f\\left( x \\right) = ;\\frac{{2\\left( {x + 4} \\right)\\left( {x - 4} \\right)}}{{\\left( {x + 4} \\right)\\left( {x + 1} \\right)}} $$\nFragments Make content appear incrementally\n{{% fragment %}} One {{% /fragment %}} {{% fragment %}} **Two** {{% /fragment %}} {{% fragment %}} Three {{% /fragment %}} Press Space to play!\nOne Two Three A fragment can accept two optional parameters:\nclass: use a custom style (requires definition in custom CSS) weight: sets the order in which a fragment appears Speaker Notes Add speaker notes to your presentation\n{{% speaker_note %}} - Only the speaker can read these notes - Press `S` key to view {{% /speaker_note %}} Press the S key to view the speaker notes!\nOnly the speaker can read these notes Press S key to view Themes black: Black background, white text, blue links (default) white: White background, black text, blue links league: Gray background, white text, blue links beige: Beige background, dark text, brown links sky: Blue background, thin dark text, blue links night: Black background, thick white text, orange links serif: Cappuccino background, gray text, brown links simple: White background, black text, blue links solarized: Cream-colored background, dark green text, blue links Custom Slide Customize the slide style and background\n{{\u0026lt; slide background-image=\u0026#34;/media/boards.jpg\u0026#34; \u0026gt;}} {{\u0026lt; slide background-color=\u0026#34;#0000FF\u0026#34; \u0026gt;}} {{\u0026lt; slide class=\u0026#34;my-style\u0026#34; \u0026gt;}} Custom CSS Example Let’s make headers navy colored.\nCreate assets/css/reveal_custom.css with:\n.reveal section h1, .reveal section h2, .reveal section h3 { color: navy; } Questions? Ask\nDocumentation\n","date":1549324800,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1549324800,"objectID":"0e6de1a61aa83269ff13324f3167c1a9","permalink":"https://frankschae.github.io/slides/example/","publishdate":"2019-02-05T00:00:00Z","relpermalink":"/slides/example/","section":"slides","summary":"An introduction to using Hugo Blox Builder's Slides feature.","tags":[],"title":"Slides","type":"slides"},{"authors":["Frank Schäfer","Robert Ford"],"categories":null,"content":" Click the Cite button above to demo the feature to enable visitors to import publication metadata into their reference management software. Create your slides in Markdown - click the Slides button to check out the example. Add the publication’s full text or supplementary notes here. You can use rich formatting such as including code, math, and images.\n","date":1441065600,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1441065600,"objectID":"966884cc0d8ac9e31fab966c4534e973","permalink":"https://frankschae.github.io/publication/journal-article/","publishdate":"2017-01-01T00:00:00Z","relpermalink":"/publication/journal-article/","section":"publication","summary":"Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum.","tags":["Source Themes"],"title":"An example journal article","type":"publication"},{"authors":["Frank Schäfer","Robert Ford"],"categories":null,"content":" Click the Cite button above to demo the feature to enable visitors to import publication metadata into their reference management software. Create your slides in Markdown - click the Slides button to check out the example. Add the publication’s full text or supplementary notes here. You can use rich formatting such as including code, math, and images.\n","date":1372636800,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":1372636800,"objectID":"69425fb10d4db090cfbd46854715582c","permalink":"https://frankschae.github.io/publication/conference-paper/","publishdate":"2017-01-01T00:00:00Z","relpermalink":"/publication/conference-paper/","section":"publication","summary":"Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum.","tags":[],"title":"An example conference paper","type":"publication"},{"authors":["FS in collaboration with Felix Eckert and Thomas Wellens"],"categories":[],"content":"We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering cross section for an incoming plane wave, on the one hand, and to minimize the decay rate associated to a long-lived scattering resonance, on the other. In both cases, the optimum is achieved by configurations where all scatterers are placed on a line parallel to the direction of the incoming plane wave. The associated maximal scattering cross section increases quadratically with the number of scatterers for large N, whereas the minimal decay rate—which is realized by configurations that are not the same as those that maximize the scattering cross section—decreases exponentially as a function of N. Finally, we also analyze the stability of our optimized configurations with respect to small random displacements of the scatterers. These results demonstrate that optimized configurations of scatterers bear a considerable potential for applications such as quantum memories or mirrors consisting of only a few atoms.\n","date":-62135596800,"expirydate":-62135596800,"kind":"page","lang":"en","lastmod":-62135596800,"objectID":"00a3646bc080207c242cecd1daa16abd","permalink":"https://frankschae.github.io/project/scattering/","publishdate":"0001-01-01T00:00:00Z","relpermalink":"/project/scattering/","section":"project","summary":"Numerical optimization of the positions of point scatterers to maximize the total scattering cross section for an incoming plane wave.","tags":["multiple scattering theory"],"title":"Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers","type":"project"}]
\ No newline at end of file
diff --git a/index.xml b/index.xml
index 8be3031..effbd20 100644
--- a/index.xml
+++ b/index.xml
@@ -5,13 +5,33 @@
     <link>https://frankschae.github.io/</link>
       <atom:link href="https://frankschae.github.io/index.xml" rel="self" type="application/rss+xml" />
     <description>FS</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Mon, 15 Jan 2024 10:49:24 -0500</lastBuildDate>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Mon, 24 Oct 2022 00:00:00 +0000</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
       <title>FS</title>
       <link>https://frankschae.github.io/</link>
     </image>
     
+    <item>
+      <title>Example Talk</title>
+      <link>https://frankschae.github.io/talk/example-talk/</link>
+      <pubDate>Sat, 01 Jun 2030 13:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/talk/example-talk/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Click on the &lt;strong&gt;Slides&lt;/strong&gt; button above to view the built-in slides feature.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Slides can be added in a few ways:&lt;/p&gt;
+&lt;ul&gt;
+&lt;li&gt;&lt;strong&gt;Create&lt;/strong&gt; slides using Hugo Blox Builder&amp;rsquo;s &lt;a href=&#34;https://docs.hugoblox.com/reference/content-types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;em&gt;Slides&lt;/em&gt;&lt;/a&gt; feature and link using &lt;code&gt;slides&lt;/code&gt; parameter in the front matter of the talk file&lt;/li&gt;
+&lt;li&gt;&lt;strong&gt;Upload&lt;/strong&gt; an existing slide deck to &lt;code&gt;static/&lt;/code&gt; and link using &lt;code&gt;url_slides&lt;/code&gt; parameter in the front matter of the talk file&lt;/li&gt;
+&lt;li&gt;&lt;strong&gt;Embed&lt;/strong&gt; your slides (e.g. Google Slides) or presentation video on this page using &lt;a href=&#34;https://docs.hugoblox.com/reference/markdown/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shortcodes&lt;/a&gt;.&lt;/li&gt;
+&lt;/ul&gt;
+&lt;p&gt;Further event details, including &lt;a href=&#34;https://docs.hugoblox.com/reference/markdown/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;page elements&lt;/a&gt; such as image galleries, can be added to the body of this page.&lt;/p&gt;
+</description>
+    </item>
+    
     <item>
       <title>Automatic Differentiation of Programs with Discrete Randomness</title>
       <link>https://frankschae.github.io/project/stochasticad/</link>
@@ -53,128 +73,91 @@
       <guid>https://frankschae.github.io/post/gsoc-2021/</guid>
       <description>&lt;h2 id=&#34;project-summary&#34;&gt;Project summary&lt;/h2&gt;
 &lt;p&gt;In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.&lt;/p&gt;
-&lt;p&gt;Possible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/projects/#5357798591823872&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the GSoC page&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Possible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available &lt;a href=&#34;https://summerofcode.withgoogle.com/projects/#5357798591823872&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the GSoC page&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;blog-posts&#34;&gt;Blog posts&lt;/h2&gt;
 &lt;p&gt;The following blog posts describe the work throughout the GSoC period in more detail:&lt;/p&gt;
 &lt;ol&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/hybridde/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Neural Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/shadowing/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/bouncing_ball/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity Analysis of Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/abstract_differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl for AD-backend agnostic code&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/hybridde/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Neural Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/shadowing/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/bouncing_ball/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity Analysis of Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/abstract_differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl for AD-backend agnostic code&lt;/a&gt;&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;docs&#34;&gt;Docs&lt;/h2&gt;
-&lt;p&gt;Documentation with respect to the adjoint sensitivity tools will be available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the local sensitivity analysis&lt;/a&gt; and 
-&lt;a href=&#34;http://scimlbase.sciml.ai/dev/fundamentals/Differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the control of automatic differentiation choices&lt;/a&gt; pages.&lt;/p&gt;
+&lt;p&gt;Documentation with respect to the adjoint sensitivity tools will be available &lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the local sensitivity analysis&lt;/a&gt; and &lt;a href=&#34;http://scimlbase.sciml.ai/dev/fundamentals/Differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the control of automatic differentiation choices&lt;/a&gt; pages.&lt;/p&gt;
 &lt;h2 id=&#34;achievements&#34;&gt;Achievements&lt;/h2&gt;
 &lt;p&gt;Below is a list of PRs in the various repositories in chronological order.&lt;/p&gt;
 &lt;h4 id=&#34;diffeqsensitivityjl&#34;&gt;DiffEqSensitivity.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/415&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add additive noise downstream test for DiffEqFlux&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/416&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallback fixes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/417&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for changes of p in callbacks&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/418&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix for using the correct uleft/pleft in continuous callback&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/419&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix broadcasting error on steady state adjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/420&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Forward Least Squares Shadowing (LSS)&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/422&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint-mode for the LSS method&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/423&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve dispatch for LSS methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/437&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Non-Intrusive Least Square Shadowing (NILSS)&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/442&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve for NILSS&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/443&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove allocation in NILSS&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/444&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Handle additional callback case&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;State-dependent Continuous Callbacks for BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/474&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;QuadratureAdjoint() for ContinuousCallback&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/475&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;More tests for Neural ODEs with callbacks for different sensitivity algorithms&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/476&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support for PeriodicCallbacks in continuous adjoint methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/415&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add additive noise downstream test for DiffEqFlux&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/416&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallback fixes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/417&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for changes of p in callbacks&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/418&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix for using the correct uleft/pleft in continuous callback&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/419&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix broadcasting error on steady state adjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/420&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Forward Least Squares Shadowing (LSS)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/422&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint-mode for the LSS method&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/423&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve dispatch for LSS methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/437&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Non-Intrusive Least Square Shadowing (NILSS)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/442&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve for NILSS&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/443&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove allocation in NILSS&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/444&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Handle additional callback case&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;State-dependent Continuous Callbacks for BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/474&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;QuadratureAdjoint() for ContinuousCallback&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/475&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;More tests for Neural ODEs with callbacks for different sensitivity algorithms&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/476&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support for PeriodicCallbacks in continuous adjoint methods&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;abstractdifferentiationjl&#34;&gt;AbstractDifferentiation.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/2&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes gradient, Jacobian, Hessian, and vjp tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/2&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes gradient, Jacobian, Hessian, and vjp tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/3&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add ForwardDiff and Zygote&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/3&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add ForwardDiff and Zygote&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;ordinarydiffeqjl&#34;&gt;OrdinaryDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/OrdinaryDiffEq.jl/pull/1424&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix discrete reverse mode for some standard controllers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/OrdinaryDiffEq.jl/pull/1424&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix discrete reverse mode for some standard controllers&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqcallbacksjl&#34;&gt;DiffEqCallbacks.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqCallbacks.jl/pull/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Introduce a PeriodicCallbackAffect struct&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqCallbacks.jl/pull/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Introduce a PeriodicCallbackAffect struct&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;steadystatediffeqjl&#34;&gt;SteadyStateDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/SteadyStateDiffEq.jl/pull/31&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Convert alg.tspan to type of prob.u0&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/SteadyStateDiffEq.jl/pull/31&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Convert alg.tspan to type of prob.u0&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;chainrulesjl&#34;&gt;ChainRules.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/506&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Do not differentiate through the construction of BitArray&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/508&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Use splatting in BitArray&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/506&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Do not differentiate through the construction of BitArray&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/508&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Use splatting in BitArray&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqnoiseprocessjl&#34;&gt;DiffEqNoiseProcess.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/94&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow solvers to use Noise Grid with SVectors&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/94&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow solvers to use Noise Grid with SVectors&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;stochasticdiffeqjl&#34;&gt;StochasticDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/428&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove Ihat2 matrix from weak solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/428&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove Ihat2 matrix from weak solvers&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqdocsjl&#34;&gt;DiffEqDocs.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/490&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Small typo on plot page&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/492&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add docs for shadowing methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/490&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Small typo on plot page&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/492&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add docs for shadowing methods&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;future-work&#34;&gt;Future work&lt;/h2&gt;
 &lt;p&gt;Besides the implementation of more shadowing methods, such as&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/1801.08674&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NILSAS&lt;/a&gt;,&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/1711.06633&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FD-NILSS&lt;/a&gt;, or&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/2009.00595&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fast linear response&lt;/a&gt;,&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/1801.08674&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NILSAS&lt;/a&gt;,&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/1711.06633&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FD-NILSS&lt;/a&gt;, or&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/2009.00595&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fast linear response&lt;/a&gt;,&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;we are planning to&lt;/p&gt;
 &lt;ul&gt;
@@ -182,20 +165,10 @@
 &lt;li&gt;refine the AbstractDifferentiation.jl package and use it within DiffEqSensitivity.jl,&lt;/li&gt;
 &lt;li&gt;add more docs and examples.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the 
-&lt;a href=&#34;https://discourse.julialang.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julia language discourse&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the &lt;a href=&#34;https://discourse.julialang.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julia language discourse&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;acknowledgement&#34;&gt;Acknowledgement&lt;/h2&gt;
-&lt;p&gt;Many thanks to my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times.
-I would also like to thank 
-&lt;a href=&#34;https://quantumtheory-bruder.physik.unibas.ch/en/people/group-members/christoph-bruder/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/arnoldjulian&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julian Arnold&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mako-git&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Martin Koppenhöfer&lt;/a&gt; for helpful comments on my blog posts. Special thanks to 
-&lt;a href=&#34;https://github.com/Zymrael&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Michael Poli&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/massastrello&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stefano Massaroli&lt;/a&gt; for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
+&lt;p&gt;Many thanks to my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times.
+I would also like to thank &lt;a href=&#34;https://quantumtheory-bruder.physik.unibas.ch/en/people/group-members/christoph-bruder/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, &lt;a href=&#34;https://github.com/arnoldjulian&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julian Arnold&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mako-git&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Martin Koppenhöfer&lt;/a&gt; for helpful comments on my blog posts. Special thanks to &lt;a href=&#34;https://github.com/Zymrael&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Michael Poli&lt;/a&gt; and &lt;a href=&#34;https://github.com/massastrello&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stefano Massaroli&lt;/a&gt; for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
 </description>
     </item>
     
@@ -204,33 +177,17 @@ I would also like to thank
       <link>https://frankschae.github.io/post/abstract_differentiation/</link>
       <pubDate>Sun, 01 Aug 2021 12:03:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/abstract_differentiation/</guid>
-      <description>&lt;p&gt;
-&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is 
-&lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&#39; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and 
-&lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, 
-&lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different 
-&lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by 
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a 
-&lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
-&lt;p&gt;
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
+      <description>&lt;p&gt;&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is &lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&amp;rsquo; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and &lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, &lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different &lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by &lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a &lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
+&lt;p&gt;&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
 &lt;h2 id=&#34;optimization-of-the-rosenbrock-function&#34;&gt;Optimization of the Rosenbrock function&lt;/h2&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
+&lt;p&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
 &lt;p&gt;$$
 g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2.
 $$&lt;/p&gt;
 &lt;p&gt;The function $g$ has a global minimum at $(x_1^\star, x_2^\star)= (a, a^2)$ with $g(x_1^\star, x_2^\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.&lt;/p&gt;
-&lt;p&gt;Let us now implement the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
+&lt;p&gt;Let us now implement the &lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
 &lt;p&gt;$$
 S({\bf x}) = \frac{1}{2} \sum_{i=1}^M f_i({\bf x})^2.
 $$&lt;/p&gt;
@@ -238,56 +195,48 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 {\bf x}^{k+1} = {\bf x}^k - \alpha_k \left(J^T J \right)^{-1} J^T f({\bf x}^k),
 $$
-where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a 
-&lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
+where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a &lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;The following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\bf x^\star}=(1,1)$ and our initial guess ${\bf x_0}=(0,-0.1)$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Pkg
-path = @__DIR__
-cd(path); Pkg.activate(&amp;quot;.&amp;quot;); Pkg.instantiate()
-
-## AbstractDifferentiation is not released yet!!
-
-using AbstractDifferentiation
-using Test, LinearAlgebra
-using FiniteDifferences, ForwardDiff, Zygote
-using Enzyme, UnPack
-using Plots, LaTeXStrings
-# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6
-
-## Rosenbrock function
-# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²
-g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2
-
-# visualization
-p = [1.0,100.0]
-x₀ = [0.0,-0.1]
-xopt = [1.0,1.0]
-
-do_plot = true
-if do_plot    
-    x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5
-    z = Surface((x₁,x₂)-&amp;gt;g([x₁,x₂],p), x₁, x₂)
-    pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true,
-                labelfontsize=20,camera = (3,50),
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
-
-    pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp),
-                labelfontsize=20,
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
-    scatter!(pl2, [(x₀[1],x₀[2])], label=L&amp;quot;x_0&amp;quot;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green)
-    scatter!(pl2, [(xopt[1],xopt[2])],label=L&amp;quot;x^\star&amp;quot;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red)
-
-    pl = plot(pl1,pl2, layout=(2,1))
-    savefig(pl, &amp;quot;Rosenbrock.png&amp;quot;)
-end
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@__DIR__&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;activate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;.&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;instantiate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## AbstractDifferentiation is not released yet!!&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AbstractDifferentiation&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Enzyme&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;UnPack&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Rosenbrock function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualization&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt;    
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.6&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.5&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;linealpha&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;colorbar&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;camera&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;heatmap&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_0&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:circle&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:green&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x^\star&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:star&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:red&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Rosenbrock.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -298,238 +247,223 @@ end
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Rosenbrock.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;To apply the Gauss-Newton algorithm to the Rosenbrock function $g$, we first cast $g$ into an appropriate form fulfilling $S({\bf x})$, i.e., we use:&lt;/p&gt;
 &lt;p&gt;$$
 f:\mathbb{R}^2\rightarrow\mathbb{R}^2:  {\bf x} \mapsto \begin{pmatrix}
-f_1({\bf x}) \\&lt;br&gt;
-f_2({\bf x}) \\&lt;br&gt;
+f_1({\bf x}) \\
+f_2({\bf x}) \\
 \end{pmatrix} = \begin{pmatrix}
-\sqrt{2}(a-x_1) \\&lt;br&gt;
-\sqrt{2b}(x_2-x_1^2)\\&lt;br&gt;
+\sqrt{2}(a-x_1) \\
+\sqrt{2b}(x_2-x_1^2)\\
 \end{pmatrix},
 $$&lt;/p&gt;
 &lt;p&gt;instead of $g$. We can easily compute the Jacobian of $f$ manually&lt;/p&gt;
 &lt;p&gt;$$
 J =  \begin{pmatrix}
--\sqrt{2} &amp;amp; 0 \\&lt;br&gt;
--2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\&lt;br&gt;
+-\sqrt{2} &amp;amp; 0 \\
+-2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\
 \end{pmatrix}.
 $$&lt;/p&gt;
 &lt;p&gt;We can then implement a (simple, non-optimized) version of the Gauss-Newton algorithm as follows.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# bring Rosenbrock function into the form &amp;quot;sum of squares of functions&amp;quot;
-f1(x,p) = convert(eltype(x),sqrt(2))*(p[1]-x[1])
-f2(x,p) = convert(eltype(x),sqrt(2*p[2]))*(x[2]-x[1]^2)
-f(x,p) = [f1(x,p),f2(x,p)]
-function f(res,x,p) # Enzyme works with inplace functions
-	res[1] = f1(x,p)
-	res[2] = f2(x,p)
-	return nothing
-end
-
-## manually pre-defined Jacobian
-function Jacobian(x,p)
-  [-convert(eltype(x),sqrt(2))   0
-  -2*x[1]*convert(eltype(x),sqrt(2*p[2]))  convert(eltype(x),sqrt(2*p[2]))]
-end
-
-## Gauss-Newton scheme
-function GaussNewton!(xs, x, p; maxiter=8, backend=nothing)
-    for i=1:maxiter
-        x = step(x, p, backend)
-        @info i
-        @show x
-        push!(xs, x)
-    end
-    return xs, x
-end
-done(x,x2,p) = g(x2,p) &amp;lt; g(x,p)
-function step(x, p, backend::Nothing, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = Jacobian(x,p)
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;xs = [x₀]
-GaussNewton!(xs, x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# output:
-[ Info: 1 ]
-x = [0.125, -0.08750000000000001]
-[ Info: 2 ]
-x = [0.234375, -0.047265625000000006]
-[ Info: 3 ]
-x = [0.4257812499999995, 0.06800537109374968]
-[ Info: 4 ]
-x = [0.5693359374999986, 0.21857223510742047]
-[ Info: 5 ]
-x = [0.784667968749996, 0.5165503501892037]
-[ Info: 6 ]
-x = [0.9999999999999961, 0.9536321163177449]
-[ Info: 7 ]
-x = [0.9999999999999989, 0.9999999999999999]
-[ Info: 8 ]
-x = [1.0, 1.0]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of 
-&lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## FiniteDifferences
-struct FDMBackend{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend() = FDMBackend(central_fdm(5, 1))
-const fdm_backend = FDMBackend()
-# Minimal interface
-AD.@primitive function jacobian(ab::FDMBackend, f, xs...)
-    return FiniteDifferences.jacobian(ab.alg, f, xs...)
-end
-
-# AD Jacobian returns tuple
-# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]
-# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]
-
-@test AD.jacobian(fdm_backend, x-&amp;gt;f(x,p), x₀)[1] ≈ Jacobian(x₀, p)
-@test AD.jacobian(fdm_backend, f, x₀, p)[1] ≈ Jacobian(x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function step(x, p, backend, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = AD.jacobian(backend, f, x, p)[1]
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-
-
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=fdm_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If we want to use reverse-mode AD instead, for example via 
-&lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Zygote
-struct ZygoteBackend &amp;lt;: AD.AbstractReverseMode end
-const zygote_backend = ZygoteBackend()
-AD.@primitive function pullback_function(ab::ZygoteBackend, f, xs...)
-    return function (vs)
-        # Supports only single output
-        _, back = Zygote.pullback(f, xs...)
-        if vs isa AbstractVector
-            back(vs)
-        else
-            @assert length(vs) == 1
-            back(vs[1])
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(zygote_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=zygote_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (
-&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## ForwardDiff
-struct ForwardDiffBackend &amp;lt;: AD.AbstractForwardMode end
-const forwarddiff_backend = ForwardDiffBackend()
-AD.@primitive function pushforward_function(ab::ForwardDiffBackend, f, xs...)
-    # jvp = f&#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0
-    return function (vs)
-        if xs isa Tuple
-            @assert length(xs) &amp;lt;= 2
-            if length(xs) == 1
-                (ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1]),0),)
-            else
-                ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1], xs[2]+h*vs[2]),0)
-            end
-        else
-            ForwardDiff.derivative(h-&amp;gt;f(xs+h*vs),0)
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(forwarddiff_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=forwarddiff_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;where we have used that the Jacobian-vector product $f&#39;(x)v$, i.e., the primitives of forward-mode AD, can be computed by 
-&lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. 
-&lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Enzyme
-struct EnzymeBackend{T1,T2,T3,T4} &amp;lt;: AD.AbstractReverseMode
-    out::T1
-    λ::T2
-    ∂f_∂x::T3
-    ∂f_∂p::T4
-end
-
-out = zero(x₀)
-λ = zero(x₀)
-∂f_∂x = zero(x₀)
-∂f_∂p = zero(p)
-
-const enzyme_backend = EnzymeBackend(out,λ,∂f_∂x,∂f_∂p)
-AD.@primitive function pullback_function(ab::EnzymeBackend, f, xs...)
-    return function (vs)  
-        # enzyme works only with inplace functions
-        if !(vs isa AbstractVector)
-            @assert length(vs) == 1 # Supports only single output
-            vs = vs[1]
-        end
-
-        if xs isa Tuple
-            @assert length(xs) == 2  # hard-coded for use case with two inputs
-            x₀ = xs[1]
-            p = xs[2]
-        end
-
-        @unpack out, λ, ∂f_∂x, ∂f_∂p = ab # cached in the struct, could also be created in here
-
-        ∂f_∂x .*= false
-        ∂f_∂p .*= false
-        out .*= false
-
-        copyto!(λ, vs)
-
-        autodiff(Duplicated(out, λ), Duplicated(x₀, ∂f_∂x), Duplicated(p, ∂f_∂p)) do _out,_x, _p
-            f(_out,_x,_p)
-        end
-        return (∂f_∂x,∂f_∂p)
-    end
-end
-AD.isinplace(ab::EnzymeBackend) = true
-AD.primalvalue(ab::EnzymeBackend, nothing, f, xs) = (f(ab.out,xs...);return ab.out)
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(enzyme_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=enzyme_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# bring Rosenbrock function into the form &amp;#34;sum of squares of functions&amp;#34;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Enzyme works with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## manually pre-defined Jacobian&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;   &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Gauss-Newton scheme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# output:&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.125&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.08750000000000001&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.234375&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.047265625000000006&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.4257812499999995&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.06800537109374968&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5693359374999986&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.21857223510742047&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.784667968749996&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5165503501892037&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;6&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999961&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9536321163177449&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;7&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999989&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9999999999999999&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of &lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## FiniteDifferences&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Minimal interface&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# AD Jacobian returns tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If we want to use reverse-mode AD instead, for example via &lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## ForwardDiff&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractForwardMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c&#34;&gt;# jvp = f&amp;#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;where we have used that the Jacobian-vector product $f&amp;rsquo;(x)v$, i.e., the primitives of forward-mode AD, can be computed by &lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. &lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Enzyme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;  
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# enzyme works only with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;  &lt;span class=&#34;c&#34;&gt;# hard-coded for use case with two inputs&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@unpack&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# cached in the struct, could also be created in here&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;autodiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;do&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;isinplace&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;primalvalue&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
 &lt;h2 id=&#34;some-current-glitches&#34;&gt;Some current glitches&lt;/h2&gt;
 &lt;p&gt;First, the push forward of a tuple of vectors, e.g., $(v_1, v_2)$, for a function with several input arguments is currently ambiguous. While &lt;code&gt;AD.jacobian&lt;/code&gt; primitives and &lt;code&gt;AD.pullback_function&lt;/code&gt; primitives interpret the push forward of our $f$ function as&lt;/p&gt;
 &lt;p&gt;$$
@@ -539,86 +473,77 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\partial f(x_0,p)}{\partial x} v_1 + \frac{\partial f(x_0,p)}{\partial p} v_2.
 $$&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous
-vs = (randn(2), randn(2))
-res1 = AD.pushforward_function(fdm_backend, f, x₀, p)(vs)
-res2 = AD.pushforward_function(forwarddiff_backend, f, x₀, p)(vs)
-
-@test res2 ≈ res1[1] + res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;res2a = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((vs[1], zero(vs[2])))
-res2b = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((zero(vs[1]), vs[2]))
-
-@test res2a ≈ res1[1]
-@test res2b ≈ res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;struct FDMBackend2{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend2() = FDMBackend2(central_fdm(5, 1))
-const fdm_backend2 = FDMBackend2()
-AD.@primitive function pushforward_function(ab::FDMBackend2, f, xs...)
-    return function (vs)
-        FDM.jvp(ab.alg, f, tuple.(xs, vs)...)
-    end
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian only defined with respect to single input variable
-@test_throws AssertionError H1 = AD.hessian(forwarddiff_backend, g, x₀, p)
-H1 = AD.hessian(forwarddiff_backend, x-&amp;gt;g(x,p), x₀)
-H2 = AD.hessian(forwarddiff_backend, p-&amp;gt;g(x₀,p), p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)
-backends = AD.HigherOrderBackend((forwarddiff_backend,zygote_backend))
-H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;FDM&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jvp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tuple&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian only defined with respect to single input variable&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test_throws&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AssertionError&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;HigherOrderBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;There are many other use cases, e.g.,&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;AbstractDifferentiation.jl is by no means complete yet. We are still in the very early stages, but we hope to make significant progress in the coming weeks. Some of the next steps are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;fixing remaining bugs, e.g., with respect to the computation of the Hessian and&lt;/li&gt;
-&lt;li&gt;adding AD/Finite Differentiation packages such as 
-&lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
+&lt;li&gt;adding AD/Finite Differentiation packages such as &lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -630,69 +555,77 @@ H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
       <description>&lt;p&gt;In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. We highlight differences between explicit&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and implicit events&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;. As a paradigmatic example, we consider a bouncing ball described by the ODE&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\&lt;br&gt;
+\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\
 \text{d}v(t) &amp;amp;= -\mathrm g\thinspace  \text{d}t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;with initial condition&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t=0) &amp;amp;= z_0 = 5, \\&lt;br&gt;
+z(t=0) &amp;amp;= z_0 = 5, \\
 v(t=0) &amp;amp;= v_0 = -0.1.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;The initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.&lt;/p&gt;
-&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
+&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
 &lt;p&gt;When ignoring the bounces, we can straightforwardly integrate the ODE analytically&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\&lt;br&gt;
+z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\
 v(t) &amp;amp;= v_0 - \mathrm g\thinspace  t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;or numerically using the OrdinaryDiffEq package from the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### simulate forward
+&lt;p&gt;or numerically using the OrdinaryDiffEq package from the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### simulate forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# dynamics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# parameters and solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;5.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.9&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward_no_bounce.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
+
+
 
-using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity
-using Plots, LaTeXStrings
 
-# dynamics
-function f(du,u,p,t)
-  du[1] = u[2]
-  du[2] = -p[1]
-end
 
-# parameters and solve
-z0 = 5.0
-v0 = -0.1
-t0 = 0.0
-tend = 1.9
-g = 10
-γ = 0.8
 
-u0 = [z0,v0]
-tspan = (t0,tend)
-p = [g, γ]
-prob = ODEProblem(f,u0,tspan,p)
 
-# plot forward trajectory
-sol = solve(prob,Tsit5(),saveat=0.1)
-pl = plot(sol, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomleft)
-hline!(pl, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34;&gt;&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
+&lt;/p&gt;
 
 
 
@@ -701,25 +634,27 @@ savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+
+
+
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward_no_bounce.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Of course, this way the ball continues to fall through the barrier at $z=0$.&lt;/p&gt;
 &lt;h2 id=&#34;forward-simulation-with-events&#34;&gt;Forward simulation with events&lt;/h2&gt;
 &lt;p&gt;At time $\tau$ around $\tau \approx 1$, the ball hits the ground $z(\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\tau$ with new initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau}) &amp;amp;=  z(\tau-) ,\\&lt;br&gt;
+z({\tau}) &amp;amp;=  z(\tau-) ,\\
 v({\tau})&amp;amp;= -\gamma v(\tau-) ,
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -736,15 +671,15 @@ $$&lt;/p&gt;
 &lt;p&gt;The full explicit trajectory $z_{\rm exp}(t) = z(t)$ is determined by&lt;/p&gt;
 &lt;p&gt;$$
 z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\&lt;br&gt;
--0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\
+-0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\
 \quad  - (-0.99005 + t) (-0.99005 \mathrm g + v_0)\gamma  ,&amp;amp; \forall t \ge \tau,
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;where we used&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\&lt;br&gt;
+z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\
 v({\tau})&amp;amp;= -\gamma v({\tau-}) = -\gamma(v_0 - 0.99005 \mathrm g)  .
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -753,15 +688,14 @@ $$&lt;/p&gt;
 a(z,v) = (z, -\gamma v).
 $$&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;DiscreteCallback&lt;/code&gt; in this case to simulate the system.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with DiscreteCallback (explicit event)
-tstar = (v0 + sqrt(v0^2+2*z0*g))/g
-condition1(u,t,integrator) = (t == tstar)
-affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2]
-cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true))
-
-sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar])
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with DiscreteCallback (explicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiscreteCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstops&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
 &lt;h3 id=&#34;implicit-events&#34;&gt;Implicit events&lt;/h3&gt;
 &lt;p&gt;The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)&lt;/p&gt;
 &lt;p&gt;$$
@@ -770,7 +704,7 @@ $$&lt;/p&gt;
 &lt;p&gt;where an event occurs at time $\tau$ if $g(z(\tau),v(\tau),p,\tau) = 0$. We have already used this condition to define our impact time $\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time  $g(z,v,p,t) = 0$.&lt;/p&gt;
 &lt;p&gt;As in the previous case, we can analytically compute the full trajectory of the ball. By substituting the formula for $\tau$ we have at the event time&lt;/p&gt;
 &lt;p&gt;\begin{aligned}
-z({\tau})&amp;amp;= 0, \\&lt;br&gt;
+z({\tau})&amp;amp;= 0, \\
 v({\tau}-)&amp;amp;= - \sqrt{v_0^2 + 2 \mathrm g z_0}
 \end{aligned}&lt;/p&gt;
 &lt;p&gt;for the left limit and&lt;/p&gt;
@@ -780,34 +714,26 @@ v({\tau})&amp;amp;= \gamma \sqrt{v_0^2 + 2 \mathrm g z_0}
 &lt;p&gt;right after the bounce. Thus, the full trajectory $z_{\rm imp}(t) = z(t)$ is given by&lt;/p&gt;
 &lt;p&gt;$$
 (\star) \quad z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\&lt;br&gt;
--\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\
+-\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\
 \quad\cdot \space (-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0} (1 + 2 \gamma)), &amp;amp; \forall t \ge \tau.
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;This is correct even if one substitutes, e.g., a value with higher precision $\mathrm g = 9.81$ for the gravitation constant.&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;ContinuousCallback&lt;/code&gt; in this case.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with ContinuousCallback (implicit event)
-condition2(u,t,integrator) = u[1] # Event happens when condition2(u,t,integrator) == 0
-cb2 = ContinuousCallback(condition2,affect!,save_positions=(true,true))
-sol2 = solve(prob,Tsit5(),callback=cb2,saveat=0.1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# plot forward trajectory
-pl1 = plot(sol1, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;explicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-pl2 = plot(sol2, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;implicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-hline!(pl1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-hline!(pl2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-pl = plot(pl1,pl2)
-savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with ContinuousCallback (implicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Event happens when condition2(u,t,integrator) == 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ContinuousCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;explicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;implicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -818,18 +744,17 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In addition, the implicitly defined impact time via the &lt;code&gt;ContinuousCallback&lt;/code&gt; also changes appropriately when changing the initial conditions or the parameters, for example when using  $\mathrm g = 9.81$ for the gravitation constant. In other words, the event time $\tau=\tau(p,z_0,v_0,t_0)$ is a function of the parameters and initial conditions, and is implicitly defined by the event condition.&lt;/p&gt;
 &lt;p&gt;Suppose we let the ball drop from a somewhat higher position now. Does an increase in height $z$ at $t=0$ give an increase or decrease in height at the end time $t_\text{end}=1.9$? This is something we can answer with sensitivity analysis. For example if we increase the height by (a fraction of) one unit then using $(\star)$&lt;/p&gt;
@@ -838,36 +763,32 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 $$&lt;/p&gt;
 &lt;p&gt;meaning the height at $t_\text{end}$ is also by a corresponding fraction of 0.84 units higher.&lt;/p&gt;
 &lt;p&gt;We can verify this visually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# animate forward trajectory
-sol3 = solve(remake(prob,u0=[u0[1]+0.5,u0[2]]),Tsit5(),callback=cb2,saveat=0.01)
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# animate forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]]),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-plt2 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;, xlims=(t0,tend))
-hline!(plt2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-plot!(plt2, sol3, tspan=(t0,tend), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)
 
-list_plots = []
-for t in sol3.t
-  tstart = 0.0
-
-  plt1 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;)
-  hline!(plt1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-  plot!(plt1, sol3, tspan=(t0,t), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-  scatter!(plt1,[t,t], sol3(t), color=[1, 2], label=false)
-  plt = plot(plt1,plt2)
-  push!(list_plots, plt)
-end
-
-plot(list_plots[100])
-
-anim = animate(list_plots,every=1)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
 
 
 
@@ -879,18 +800,14 @@ anim = animate(list_plots,every=1)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;The original curve is shown in black in the figure above.&lt;/p&gt;
 &lt;h2 id=&#34;sensitivity-analysis-with-events&#34;&gt;Sensitivity analysis with events&lt;/h2&gt;
@@ -905,7 +822,7 @@ $$&lt;/p&gt;
 &lt;p&gt;Inserting our results for $z(t) = z_{\rm exp}(t)$ instead of $z(t) = z_{\rm imp}(t)$ at $t_{\text{end}}$, a different value for the sensitivity is obtained (a value ignoring the changes in $z$ due to changes in the bouncing time $\tau$), e.g.,&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L^{\text{exp}}}{\text{d} z_0} = \begin{cases}
-1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\&lt;br&gt;
+1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\
 0.84 &amp;amp; \text{with } z(t) = z_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -916,7 +833,7 @@ $$&lt;/p&gt;
 &lt;p&gt;we could be interested in the sensitivity of $L$ with respect $g$. Using the velocity $v(t)$ at the explicit time $t = 0.9905$ instead of the implicit $v(\tau-)$, again a different value for the sensitivity is obtained (a value again ignoring the changes in $\tau$ and not the one we are looking for), even though $\tau = 0.9905$ in both cases:&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L}{\text{d} g} = \begin{cases}
--0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\&lt;br&gt;
+-0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\
 -\frac{g}{\sqrt{v_0^2 + 2 g z_0}} = -0.99995 &amp;amp; \text{ with } v(t) = v_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -933,51 +850,48 @@ $$&lt;/p&gt;
 &lt;p&gt;We write&lt;/p&gt;
 &lt;p&gt;$$\text{solve}(t_0, x_0, t, p)$$&lt;/p&gt;
 &lt;p&gt;$(= x(t))$ for the functional solution of the ODE at time $t$.&lt;/p&gt;
-&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
+&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
 &lt;p&gt;Let us focus on the &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; algorithm which computes the sensitivities&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\&lt;br&gt;
+\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\
 \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}p} &amp;amp;= \lambda_{p}(t_{0}),
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;of a loss function $L$ acting on the final state with respect to the initial state and the parameters. It does so by solving an ODE for $\lambda(s)$ in reverse time from $t$ to $t_0$&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\&lt;br&gt;
+\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\
 \frac{\text{d}\lambda_{p}(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(x(s), \rightarrow p, s)}{\text{d} p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;with initial conditions:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\&lt;br&gt;
+\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\
 \lambda_{p}(t) &amp;amp;= 0.
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The arrows ($\rightarrow$) indicate the variable with respect to which we differentiate, which will become important later when the same variable shows up in multiple function arguments.&lt;/p&gt;
-&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in &lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;&lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, essentially the custom primitive differentiation rule of &lt;code&gt;solve&lt;/code&gt;, is the elementary building block needed to derive sensitivities also in more complicated examples:&lt;/p&gt;
 &lt;p&gt;Consider a loss depending on the state $x(s_i)$ at fix time points $s_i$ through loss functions $L_i^{\text{exp}}$,&lt;/p&gt;
 &lt;p&gt;$$
 L^{\text{exp}} = \sum_i L_i^{\text{exp}}(s_i, x(s_i), p).
 $$&lt;/p&gt;
 &lt;p&gt;Without contortions we can obtain the sensitivity of $L^{\text{exp}}$ in $p$ (or in $x_0$) using the tool we have. For those a bit familiar with automatic differentiation, this is perhaps easiest to see if we write $L^{\text{exp}}$ as pseudo code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1 = solve(t0, x0, s1, p)
-    L = L1(s1, x1, p)
-    x2 = solve(s1, x1, s2, p)
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\&lt;br&gt;
+\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\
 \frac{\text{d}\lambda_{p}(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(x(t), \rightarrow p, t)}{\text{d} p} - \frac{\text{d} L_i^{\text{exp}}(  x(t),\rightarrow p)}{\text{d} p}^\dagger \delta(t-s_i).
 \end{aligned}
 $$&lt;/p&gt;
@@ -991,24 +905,23 @@ $$&lt;/p&gt;
 $$&lt;/p&gt;
 &lt;p&gt;Now we can even properly define the &lt;code&gt;rrule&lt;/code&gt; of &lt;code&gt;solve&lt;/code&gt; in the sense of &lt;code&gt;DiffRules.jl&lt;/code&gt;.&lt;/p&gt;
 &lt;h3 id=&#34;explicit-events-1&#34;&gt;Explicit events&lt;/h3&gt;
-&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1- = solve(t0, x0, s1, p)
-    L = L1-(s1, x1-, p) # saved before affect
-    x1 = a(x1-, p, s1) # affect
-    L += L1(s1, x1, p) # saved after affect
-    x2- = solve(s1, x1, s2, p)
-    L += L2-(s2, x2-, p)
-    x2 = a(x2-, p, s2) # affect
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
+&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref2:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved before affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved after affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\&lt;br&gt;
+\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\
 \lambda_p({s_j}-) &amp;amp;= \lambda_p({s_j}) -  \lambda({s_j})^\dagger \frac{\text{d} a(x({s_j}-), \rightarrow p, {s_j}-)}{\text{d} p}
 \end{aligned}
 $$&lt;/p&gt;
@@ -1022,27 +935,26 @@ $$&lt;/p&gt;
 &lt;p&gt;to put emphasis on this, or equivalently, compute for a unspecified function $L(t, x, p)$ the result of $\frac{\text{d}L}{\text{d}p}$ with&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\&lt;br&gt;
+\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\
 &amp;amp;= \frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;which indicates that changing $p$ influences $L$ both through changes in $\tau_1$ as well as changes in&lt;/p&gt;
 &lt;p&gt;$$\xi_1 = x(\tau_1-).$$&lt;/p&gt;
 &lt;p&gt;This case where we have a loss function $L = L_1$ depending on $\tau_1$, $x(\tau_1)$, and $p$
-was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;p&gt;Therefore, the sensitivity of the event time with respect to parameters $\frac{\text{d}\tau}{\text{d}p}$ must be taken into account.
-Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the 
-&lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
+Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the &lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\&lt;br&gt;
+\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The total derivative&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; inside the bracket is:
 $$
 \begin{aligned}
-\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\&lt;br&gt;
+\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;Since&lt;/p&gt;
@@ -1065,8 +977,8 @@ for the second term of $\dfrac{\text{d}\tau_1(p)}{\text{d}p}$.&lt;/p&gt;
 &lt;p&gt;We can now write the gradient as:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\&lt;br&gt;
-+&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\&lt;br&gt;
+\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\
++&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\
 +&amp;amp; \frac{\text{d}L(\rightarrow \tau_1(p), \text{solve}(t_0, x_0, \rightarrow \tau_1(p),  p), p)}{\text{d}\tau_1} \frac{\text{d} \tau_1(p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
@@ -1135,7 +1047,7 @@ $$&lt;/p&gt;
 &lt;li&gt;If both limits contribute to the loss function, the contributions ${\color{red}\lambda_1}$ and ${\color{blue}\lambda_{a,1}}$ are added.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h4 id=&#34;generalization-several-events&#34;&gt;Generalization: several events&lt;/h4&gt;
-&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
+&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
 &lt;p&gt;$$
 \text{solve}(t_0, x_0, t, p)  = \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p),
 $$&lt;/p&gt;
@@ -1143,16 +1055,15 @@ $$&lt;/p&gt;
 &lt;p&gt;(A good exercise to get familiar with these type of arguments is to verify&lt;/p&gt;
 &lt;p&gt;$$\frac{\text{d}}{\text{d}s} \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p) = 0.)$$&lt;/p&gt;
 &lt;p&gt;Essentially, we know how to address several events already by considering a loss function&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss2(t0, x0, p)
-    tau1, xi1 = solve2(t0, x0, g, p)
-    L = L1(tau1, xi1, p)
-    tau2, xi2 = solve2(tau1, xi1, g, p)
-    L += L2(tau2, xi2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
 &lt;p&gt;Differentiating by hand, we have the following modification of the method from the previous section, which can be derived choosing a particular function $L$ in the previous section which incorporates the &lt;code&gt;solve&lt;/code&gt; from $\tau_1$ to $t_{\text{end}}$. Note&lt;/p&gt;
 &lt;p&gt;$$
 \lambda_{\tau_1}^0 = \frac{\text{d}(\text{solve}(\rightarrow \tau_1,  a(\xi_1,p), t_{\text{end}}, p))}{\text{d} \tau_1} = - f(a(\xi_1,p), p, \tau_1).
@@ -1184,43 +1095,38 @@ $$&lt;/p&gt;
 &lt;li&gt;Backpropagate $\lambda_J = {\color{red}\lambda_J} + {\color{blue}\lambda_{a,J}} + {\color{green}\lambda_{c,J}}$ to the next event time $\tau_{J-1}$ and iterate over the remaining $J-1$ events.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
-&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
-&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (
-&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$. &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$.&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the &lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
 $$
 \begin{aligned}
-\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\&lt;br&gt;
-&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\&lt;br&gt;
+\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\
+&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\
 &amp;amp;+ \frac{\text{d}f(t, x_1(t),x_2(\rightarrow t),\dots,x_N(t))}{\text{d}t} + \dots +  \frac{\text{d}f(t, x_1(t),x_2(t),\dots,x_N(\rightarrow t))}{\text{d}t}.
 \end{aligned}
-$$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+$$&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1230,8 +1136,7 @@ $$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-
       <pubDate>Fri, 02 Jul 2021 11:08:22 +0200</pubDate>
       <guid>https://frankschae.github.io/post/shadowing/</guid>
       <description>&lt;p&gt;In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &amp;ldquo;butterfly effect&amp;rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.&lt;/p&gt;
-&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
+&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:&lt;/p&gt;
 &lt;p&gt;$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -1245,58 +1150,51 @@ Under the assumption of ergodicity, $\langle g \rangle_∞$ only depends on $p$.
 &lt;p&gt;One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{pmatrix}
-\text{d}x \\&lt;br&gt;
-\text{d}y \\&lt;br&gt;
-\text{d}z \\&lt;br&gt;
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\&lt;br&gt;
-x(\rho-z) - y\\&lt;br&gt;
-x y - \beta z \\&lt;br&gt;
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
-
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
-
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &amp;quot;initial&amp;quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&amp;quot;attractor&amp;quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Statistics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;QuadGK&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SparseArrays&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# simulate 1 trajectory of the Lorenz system forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;28.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;initial&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;attractor&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -1307,83 +1205,76 @@ savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.&lt;/p&gt;
-&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
+&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
 &lt;p&gt;$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function compute_objective(sol)
-  quadgk(t-&amp;gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&amp;quot;z(t)&amp;quot;, xlabel=L&amp;quot;t&amp;quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&amp;quot;\langle z\rangle&amp;quot;, lw = 2)
-savefig(pl2, &amp;quot;zsingle.png&amp;quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
-
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
-
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&amp;quot;\langle z\rangle&amp;quot;, xlabel=L&amp;quot;\rho&amp;quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &amp;quot;zvsrho.png&amp;quot;)
-
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &amp;quot;z.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We obtain:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;quadgk&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;atol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;mean_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zsingle.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# for each value of the parameter, solve 20 times the initial value problem&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# wrap the procedure inside a function depending on p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ribbon&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;ho&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zvsrho.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;margin&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;600&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;300&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;z.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We obtain:&lt;/p&gt;
 
 
 
 
 
-  
-
-
-
-
 
 
 
@@ -1391,40 +1282,36 @@ savefig(pl4, &amp;quot;z.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/z.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/z.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;z.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$&lt;/p&gt;
 &lt;h2 id=&#34;conventional-forward-mode-sensitivity-analysis-and-finite-differencing&#34;&gt;Conventional forward-mode sensitivity analysis and finite-differencing&lt;/h2&gt;
-&lt;p&gt;For non-chaotic systems, we would just use the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we find diverging values:&lt;/p&gt;
+&lt;p&gt;For non-chaotic systems, we would just use the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_forward&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_calculus&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we find diverging values:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\&lt;br&gt;
+&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$&lt;/p&gt;
@@ -1432,45 +1319,44 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$&lt;/p&gt;
-&lt;p&gt;Similarly, using 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-     xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&amp;quot;Float32&amp;quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
+&lt;p&gt;Similarly, using &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;500&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_Floats.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&amp;quot;Float32&amp;quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
 
-savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
-
 
 
 
@@ -1481,18 +1367,10 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Without animation:&lt;/p&gt;
 
@@ -1500,9 +1378,6 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-  
-
-
 
 
 
@@ -1512,60 +1387,52 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_Floats.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;Luckily, the 
-&lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
+&lt;p&gt;Luckily, the &lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
 &lt;blockquote&gt;
 &lt;p&gt;Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&amp;ldquo;shadows&amp;rdquo;) the numerically computed one.&lt;/p&gt;
 &lt;/blockquote&gt;
 &lt;h2 id=&#34;shadowing-methods&#34;&gt;Shadowing methods&lt;/h2&gt;
-&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref2:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref3:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;lss-and-nilss-for-the-lorenz-system&#34;&gt;LSS and NILSS for the Lorenz system&lt;/h2&gt;
-&lt;p&gt;Switching to LSS or NILSS within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
+&lt;p&gt;Switching to LSS or NILSS within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# objective&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# LSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;CosWindowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0095888187322035&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Cos2Windowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0343951385924328&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0284286902740765&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_adjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.028428690274077&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# select via sensealg in solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9694728321500617&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;&lt;code&gt;CosWindowing()&lt;/code&gt; (default)&lt;/li&gt;
 &lt;li&gt;&lt;code&gt;Cos2Windowing()&lt;/code&gt;&lt;/li&gt;
@@ -1574,72 +1441,68 @@ dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
 &lt;p&gt;Additionally, an adjoint implementation &lt;code&gt;AdjointLSS()&lt;/code&gt; is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that &lt;code&gt;AdjointLSS(alpha=10.0)&lt;/code&gt; agrees perfectly with &lt;code&gt;ForwardLSS(alpha=10.0)&lt;/code&gt;. In all cases considered, we find the expected sensitivity value of $\approx 1$.&lt;/p&gt;
 &lt;p&gt;However, the use of &lt;code&gt;LSS()&lt;/code&gt; is (typically) much more expensive than the use of &lt;code&gt;NILSS()&lt;/code&gt;, because &lt;code&gt;LSS()&lt;/code&gt; needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of &lt;code&gt;NILSS()&lt;/code&gt; scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making &lt;code&gt;NILSS()&lt;/code&gt; more efficient.&lt;/p&gt;
 &lt;p&gt;In the &lt;code&gt;NILSS()&lt;/code&gt; algorithm, the user can control the number of steps per segment as well as the number of segments.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# NILSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# make sure trajectory is fully on the attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;120.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9966924374966089&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0416028730638789&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Warning: Detected a large value of ξ at the beginning of a segment.&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref4:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;NILSAS&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;li&gt;FD-NILSS&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref4:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019) &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019)&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019) &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019)&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1648,21 +1511,11 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
       <link>https://frankschae.github.io/post/hybridde/</link>
       <pubDate>Wed, 16 Jun 2021 14:50:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/hybridde/</guid>
-      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
+      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
 &lt;p&gt;This first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).&lt;/p&gt;
-&lt;p&gt;Note: Please check the 
-&lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
+&lt;p&gt;Note: Please check the &lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
 &lt;h2 id=&#34;hybrid-differential-equations&#34;&gt;Hybrid Differential Equations&lt;/h2&gt;
-&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
+&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
 &lt;p&gt;Some relevant examples for hybrid DEs with discrete or continuous callbacks are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; (&lt;code&gt;ContinuousCallback&lt;/code&gt;).&lt;/li&gt;
@@ -1672,7 +1525,7 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
 &lt;li&gt;pharmacokinetic models&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, where explicit dosing times change the drug concentration in the blood (&lt;code&gt;DiscreteCallback&lt;/code&gt;). The simplest possible example being the one-compartment model.&lt;/li&gt;
 &lt;li&gt;kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (&lt;code&gt;DiscreteCallback&lt;/code&gt;).&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;kicked-harmonic-oscillator&#34;&gt;Kicked harmonic oscillator&lt;/h2&gt;
 &lt;p&gt;Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form&lt;/p&gt;
 &lt;p&gt;$$
@@ -1681,43 +1534,40 @@ $$&lt;/p&gt;
 &lt;p&gt;where $a=0.1$ and $b=1$ with initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-x(t=0) &amp;amp;= 1  \\&lt;br&gt;
+x(t=0) &amp;amp;= 1  \\
 v(t=0) &amp;amp;= \dot{x}(t=0) = 0.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;This second-order ODE can be 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use 
-&lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity
-using Zygote
-using Random
-u0 = Float32[1.; 0.]
-
-tspan = (0.0f0,50.0f0)
-
-dtsave = 0.5f0
-t = tspan[1]:dtsave:tspan[2]
+&lt;p&gt;This second-order ODE can be &lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use &lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqFlux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Optim&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# ODE without kicks&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-function oscillator!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  return nothing
-end
 
-prob_data = ODEProblem(oscillator!,u0,tspan)
 
-# ODE without kicks
-pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
-
-
-
 
 
 
@@ -1726,45 +1576,36 @@ pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &am
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as &lt;code&gt;1&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;kicktimes = tspan[1]:1:tspan[2]
-function kick!(integrator)
-  integrator.u[end] += one(eltype(integrator.u))
-end
-cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false))
-
-sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t)
-t_data = sol_data.t
-ode_data = Array(sol_data)
-
-# visualize data
-pl1 = plot(t_data,ode_data[1,:],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl1,t_data,ode_data[2,:],label=&amp;quot;data v(t)&amp;quot;)
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualize data&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
-pl2 = plot(t_data[1:20],ode_data[1,1:20],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl2,t_data[1:20],ode_data[2,1:20],label=&amp;quot;data v(t)&amp;quot;)
-pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;
 
 
 
 
-  
-
 
 
 
@@ -1775,64 +1616,50 @@ pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting &lt;code&gt;save_positions=(true,true)&lt;/code&gt;, the kicks would be saved before &lt;strong&gt;and&lt;/strong&gt; after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;, we now aim at learning (potentially) missing parts of the model from these data traces.&lt;/p&gt;
 &lt;h3 id=&#34;high-domain-knowledge&#34;&gt;High domain knowledge&lt;/h3&gt;
 &lt;p&gt;For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the &lt;code&gt;affect!&lt;/code&gt; function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of &lt;code&gt;integrator.u&lt;/code&gt;. If we further set the initial value of that component to &lt;code&gt;1&lt;/code&gt;, then the neural network only has to learn that &lt;code&gt;du[3]&lt;/code&gt; is &lt;code&gt;0&lt;/code&gt;. In other words, the output of the neural network must be &lt;code&gt;0&lt;/code&gt; for all states &lt;code&gt;u&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;Random.seed!(123)
-nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1))
-p_nn1 = initial_params(nn1)
-
-function f1!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3] = nn1(u[1:2], p)[1]
-  return nothing
-end
-
-affect!(integrator) = integrator.u[2] += integrator.u[3]
-cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false))
-z0 = Float32[u0;one(u0[1])]
-prob1 = ODEProblem(f1!,z0,tspan,p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# to visualize the predictions of the trained neural network below
-function visualize(prob,p)
-  _prob = remake(prob,p=p)
-  ode_pred = Array(solve(_prob,Tsit5(),callback=cb,
-                 saveat=dtsave))[1:2,:]
-  pl1 = plot(t_data,ode_pred[1,:],label=&amp;quot;x(t)&amp;quot;)
-  scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=&amp;quot;data x(t)&amp;quot;)
-  pl2 = plot(t_data,ode_pred[2,:],label=&amp;quot;v(t)&amp;quot;)
-  scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=&amp;quot;data v(t)&amp;quot;)
-
-  pl = plot(pl1, pl2, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-  return pl, sum(abs2,ode_data .- ode_pred)
-end
-
-pl = plot(solve(prob1,Tsit5(),saveat=t,
-  callback=cb
-  ),label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;123&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FastChain&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tanh&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial_params&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# to visualize the predictions of the trained neural network below&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                 &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -1843,77 +1670,66 @@ pl = plot(solve(prob1,Tsit5(),saveat=t,
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;untrained_nn.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;which (of course) doesn&amp;rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### loss function
-function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint())
-  _prob = remake(prob,p=p)
-  pred = Array(solve(_prob,Tsit5(),callback=cb,
-               saveat=dtsave,sensealg=sensealg))[1:2,:]
-  sum(abs2,ode_data .- pred)
-end
-
-loss(p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The recently implemented tools are deeply hidden within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., 
-&lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;###################################
-# training loop
-# optimize the parameters for a few epochs with ADAM
-function train(prob, p_nn; sensealg=BacksolveAdjoint())
-  opt = ADAM(0.0003f0)
-  list_plots = []
-  losses = []
-  for epoch in 1:200
-    println(&amp;quot;epoch: $epoch / 200&amp;quot;)
-    _dy, back = Zygote.pullback(p -&amp;gt; loss(p,
-      prob=prob,
-      sensealg=sensealg), p_nn)
-    gs = @time back(one(_dy))[1]
-    push!(losses, _dy)
-    if epoch % 10 == 0
-      # plot every xth epoch
-      pl, test_loss = visualize(prob, p_nn)
-      println(&amp;quot;Loss (epoch: $epoch): $test_loss&amp;quot;)
-      display(pl)
-      push!(list_plots, pl)
-    end
-    Flux.Optimise.update!(opt, p_nn, gs)
-    println(&amp;quot;&amp;quot;)
-  end
-  return losses, list_plots
-end
-
-# plot training loss
-losses, list_plots = train(prob1, p_nn1)
-pl1 = plot(losses, lw = 1.5, xlabel = &amp;quot;epoch&amp;quot;, ylabel=&amp;quot;loss&amp;quot;, legend=false)
-pl2 = list_plots[end]
-pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t,
-   callback=cb
-  ), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-
-pl = plot(pl2,pl3)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### loss function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ReverseDiffAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;               &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The recently implemented tools are deeply hidden within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., &lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;###################################&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# training loop&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# optimize the parameters for a few epochs with ADAM&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ADAM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0003f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;200&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt; / 200&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;%&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;c&#34;&gt;# plot every xth epoch&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_loss&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Loss (epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;): &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$test_loss&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;display&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Optimise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;update!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot training loss&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;epoch&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;loss&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -1924,98 +1740,91 @@ pl = plot(pl2,pl3)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/trained1.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/trained1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;trained1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We see the expected constant value of &lt;code&gt;u[3]&lt;/code&gt;, indicating a kick to the velocity of &lt;code&gt;+=1&lt;/code&gt;, at the kicking times over the full time interval.&lt;/p&gt;
 &lt;h2 id=&#34;reducing-the-domain-knowledge&#34;&gt;Reducing the domain knowledge&lt;/h2&gt;
-&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
+&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;changing the initial condition of the third component of &lt;code&gt;u&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;using another affect function &lt;code&gt;affect!(integrator) = integrator.u[2] = integrator.u[3]&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;dropping the knowledge that only &lt;code&gt;u[2]&lt;/code&gt; gets a kick by using a neural network with &lt;code&gt;2&lt;/code&gt; outputs (+ a fourth component in the ODE):&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4]
-function f2!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;fitting the parameters $a$ and $b$ simultaneously:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f3!(du,u,p,t)
-  a = p[end-1]
-  b = p[end]
-  nn_weights = p[1:end-2]
-
-  du[1] = u[2]
-  du[2] = -b*u[1] - a*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f3!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;inferring the entire underlying dynamics using a neural network with &lt;code&gt;4&lt;/code&gt; outputs:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f4!(du,u,p,t)
-  Ω = nn3(u[1:2], p)
-
-  du[1] = Ω[1]
-  du[2] = Ω[2]
-  du[3:4] .= Ω[3:4]
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f4!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;etc.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;refine the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;) and&lt;/li&gt;
-&lt;li&gt;support direct usage through the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
+&lt;li&gt;support direct usage through the &lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021) &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021)&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -2076,152 +1885,100 @@ end
       <pubDate>Wed, 26 Aug 2020 22:59:49 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc-2020/</guid>
       <description>&lt;h2 id=&#34;project-summary&#34;&gt;Project summary&lt;/h2&gt;
-&lt;p&gt;In this project, we have implemented new promising tools within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
+&lt;p&gt;In this project, we have implemented new promising tools within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
 The high weak order solvers will allow for massive performance advantages for fitting expectations of equations.
 Instead of automatic differentiation (AD) through the operations of an SDE solver, which scales poorly in memory, one can now use efficient stochastic adjoint sensitivity methods.&lt;/p&gt;
 &lt;h2 id=&#34;blog-posts&#34;&gt;Blog posts&lt;/h2&gt;
 &lt;p&gt;The following posts describe the work during the entire period in more detail:&lt;/p&gt;
 &lt;ol&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GSoC 2020: High weak order SDE solvers and their utility in neural SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/high-weak/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;High weak order SDE solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GSoC 2020: High weak order SDE solvers and their utility in neural SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/high-weak/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;High weak order SDE solvers&lt;/a&gt;&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;docs&#34;&gt;Docs&lt;/h2&gt;
-&lt;p&gt;The documentation of the solvers is available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/solvers/sde_solve/#High-Weak-Order-Methods&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.
-Docs with respect to the adjoint sensitivity tools will be available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The documentation of the solvers is available &lt;a href=&#34;https://diffeq.sciml.ai/latest/solvers/sde_solve/#High-Weak-Order-Methods&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.
+Docs with respect to the adjoint sensitivity tools will be available &lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;achievements&#34;&gt;Achievements&lt;/h2&gt;
 &lt;p&gt;Please find below a list of the PRs carried out during GSoC in the different repositories in chronological order within the SciML ecosystem.&lt;/p&gt;
 &lt;h4 id=&#34;stochasticdiffeqjl&#34;&gt;StochasticDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/285&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Inplace version of DRI1 scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/289&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI1 method&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;tstop fixes for reverse time propagation&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes for EulerHeun scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/327&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Speed up the tests for the DRI1 and RI1 schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/328&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/329&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RDI1WM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/332&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adaptive version of the stochastic Runge-Kutta schemes by embedding&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/333&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RS1 and RS2 schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/334&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PL1WM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/337&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NON scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/338&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for weak methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/342&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stochastic improved and modified Euler methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/343&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;COM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/348&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Computationally more efficient NON variant (NON2)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/285&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Inplace version of DRI1 scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/289&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI1 method&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;tstop fixes for reverse time propagation&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes for EulerHeun scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/327&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Speed up the tests for the DRI1 and RI1 schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/328&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/329&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RDI1WM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/332&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adaptive version of the stochastic Runge-Kutta schemes by embedding&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/333&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RS1 and RS2 schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/334&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PL1WM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/337&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NON scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/338&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for weak methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/342&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stochastic improved and modified Euler methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/343&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;COM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/348&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Computationally more efficient NON variant (NON2)&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/288&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Static array tests&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Levy area for non-commutative noise processes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/288&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Static array tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Levy area for non-commutative noise processes&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqsensitivityjl&#34;&gt;DiffEqSensitivity.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/235&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint sensitivities for steady states&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/237&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Concrete_solve dispatch for steady state problem&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/242&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;BacksolveAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/256&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GPU savety for SDE BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/258&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Tests for concrete solve with respect to SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/260&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Alternative differentiation choices (vjps) for noise Jacobian&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/265&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes and tests for inplace formulation of BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/268&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Efficient SDE BacksolveAdjoint for scalar noise&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/275&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/295&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;InterpolatingAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/298&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for backsolve, steadystate and interpolation adjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/299&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for more general noise processes: replace NoiseGrid by NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/303&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Cheaper non-diagonal noise tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/235&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint sensitivities for steady states&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/237&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Concrete_solve dispatch for steady state problem&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/242&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;BacksolveAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/256&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GPU savety for SDE BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/258&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Tests for concrete solve with respect to SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/260&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Alternative differentiation choices (vjps) for noise Jacobian&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/265&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes and tests for inplace formulation of BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/268&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Efficient SDE BacksolveAdjoint for scalar noise&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/275&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/295&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;InterpolatingAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/298&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for backsolve, steadystate and interpolation adjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/299&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for more general noise processes: replace NoiseGrid by NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/303&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Cheaper non-diagonal noise tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support adjoints for SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support adjoints for SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqnoiseprocessjl&#34;&gt;DiffEqNoiseProcess.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/48&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Multi-dimensional Brownian motion tests&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/49&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Bug fix for inplace form of NoiseGrid&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/51&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversible NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/53&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Relax the size constraints of the available noise processes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/54&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the real-valued white noise process function&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/55&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix of an extraction issue with NoiseGrid&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Extraction and endpoint fixes on NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversal of SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/48&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Multi-dimensional Brownian motion tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/49&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Bug fix for inplace form of NoiseGrid&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/51&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversible NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/53&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Relax the size constraints of the available noise processes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/54&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the real-valued white noise process function&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/55&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix of an extraction issue with NoiseGrid&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Extraction and endpoint fixes on NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversal of SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqgpujl&#34;&gt;DiffEqGPU.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl/pull/59&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Memory efficient reduction function for ensemble problems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl/pull/59&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Memory efficient reduction function for ensemble problems&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;modelingtoolkitjl&#34;&gt;ModelingToolkit.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqdevtoolsjl&#34;&gt;DiffEqDevTools.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NoiseWrapper alternative for analyticless convergence tests of SDE solvers&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/74&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;test_convergence() dispatch for ensemble simulations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Work precision set for ensemble problems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NoiseWrapper alternative for analyticless convergence tests of SDE solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/74&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;test_convergence() dispatch for ensemble simulations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Work precision set for ensemble problems&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqbasejl&#34;&gt;DiffEqBase.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqBase.jl/pull/503&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix concrete_solve tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqBase.jl/pull/503&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix concrete_solve tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;future-work&#34;&gt;Future work&lt;/h2&gt;
 &lt;p&gt;There is still a lot that we&amp;rsquo;d like to do, e.g.,&lt;/p&gt;
@@ -2235,12 +1992,8 @@ Docs with respect to the adjoint sensitivity tools will be available
 &lt;/ul&gt;
 &lt;p&gt;Contributions, suggestions &amp;amp; comments are always welcome! You might like to join our slac channels #diffeq-bridged and #neuralsde to get in touch.&lt;/p&gt;
 &lt;h2 id=&#34;acknowledgement&#34;&gt;Acknowledgement&lt;/h2&gt;
-&lt;p&gt;I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback.
-I would also like to thank 
-&lt;a href=&#34;https://www.quantumtheory-bruder.physik.unibas.ch/people.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts.
+&lt;p&gt;I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback.
+I would also like to thank &lt;a href=&#34;https://www.quantumtheory-bruder.physik.unibas.ch/people.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts.
 Many thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
 </description>
     </item>
@@ -2250,11 +2003,8 @@ Many thanks to the very supportive julia community and to Google&amp;rsquo;s ope
       <link>https://frankschae.github.io/post/high-weak/</link>
       <pubDate>Mon, 17 Aug 2020 14:46:35 +0200</pubDate>
       <guid>https://frankschae.github.io/post/high-weak/</guid>
-      <description>&lt;p&gt;This post summarizes our new 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
-for the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
+      <description>&lt;p&gt;This post summarizes our new &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
+for the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
 &lt;p&gt;After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
 Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics.
 Finally, we discuss how to run simulations on GPU hardware.&lt;/p&gt;
@@ -2285,16 +2035,14 @@ $$
 \rm{E}(|X(T) -Y(T)|) \leq C \cdot h^p,
 $$
 for each $h \in [0, \delta]$.&lt;/p&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
+&lt;p&gt;The &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
 for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms.
 For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI)
 can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where &lt;em&gt;b&lt;/em&gt; has only entries on its diagonal or $m=1$.
 The main difficulty for the construction of strong methods with an order &amp;gt; 1/2 arises from the need of an accurate estimation of
 multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of &lt;em&gt;dW&lt;/em&gt; in the case
 of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed
-in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also 
-&lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
+in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also &lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;weak-convergence&#34;&gt;Weak convergence&lt;/h2&gt;
 &lt;p&gt;Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt; in many situations. In these cases, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient and &amp;ndash; due to the less restrictive formulation of the objective &amp;ndash; those solvers are computationally cheaper than their strong counterparts.
 For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics.
@@ -2309,19 +2057,17 @@ $~$&lt;/p&gt;
 &lt;p&gt;for any polynomial $g$ for each $h \in [0, \delta]$.&lt;/p&gt;
 &lt;p&gt;We demonstrate below that &lt;strong&gt;high weak order solvers&lt;/strong&gt; are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.&lt;/p&gt;
 &lt;h2 id=&#34;new-high-weak-order-methods&#34;&gt;New high weak order methods&lt;/h2&gt;
-&lt;p&gt;A list of all new weak solvers is available in the 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
+&lt;p&gt;A list of all new weak solvers is available in the &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
 Note that we also implemented methods designed for the Stratonovich sense.
 For the subsequent examples regarding Ito SDEs, we use only a subset of the plethora of second-order weak solvers.
-We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
-We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
+We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
+We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref2:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref3:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
 The latter is the simplest weak solver, where the Gaussian increments of the strong Euler-Maruyama scheme are replaced by
 two-point distributed random variables with similar moment properties.&lt;/p&gt;
 &lt;p&gt;Rößler&amp;rsquo;s SRK schemes are particularly designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;, since only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn and since the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.
 &lt;code&gt;PL1WM()&lt;/code&gt; in contrast needs to simulate &lt;code&gt;m(m+1)/2&lt;/code&gt; random variables but a smaller number of order conditions needs to be fulfilled.&lt;/p&gt;
 &lt;h3 id=&#34;convergence-tests&#34;&gt;Convergence tests&lt;/h3&gt;
-&lt;p&gt;As in the 
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As in the &lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -2332,86 +2078,78 @@ $$ ~$$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;for the function $f(x)=(x_1)^2$, which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e6&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-using Test
-using Random
-using Plots
-using DiffEqDevTools
-
-function prob_func(prob, i, repeat)
-    remake(prob,seed=seeds[i])
-end
-
-u₀ = [1.0,1.0]
-function f1!(du,u,p,t)
-  @inbounds begin
-    du[1] = -273//512*u[1]
-    du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-  end
-  return nothing
-end
-function g1!(du,u,p,t)
-  @inbounds begin
-    du[1,1] = 1//4*u[1]
-    du[1,2] = 1//16*u[1]
-    du[2,1] = (1-2*sqrt(2))/4*u[1]
-    du[2,2] = 1//10*u[1]+1//16*u[2]
-  end
-  return nothing
-end
-dts = 1 .//2 .^(3: -1:0)
-tspan = (0.0,3.0)
-
-h2(z) = z^2 # but apply it only to u[1]
-
-prob = SDEProblem(f1!,g1!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-numtraj = Int(1e6)
-seed = 100
-Random.seed!(seed)
-seeds = rand(UInt, numtraj)
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h2(sol[end][1]),false),
-        prob_func = prob_func
-        )
-
-sim = test_convergence(dts,ensemble_prob,DRI1(),
-  save_everystep=false,trajectories=numtraj,save_start=false,adaptive=false,
-  weak_timeseries_errors=false,weak_dense_errors=false,
-  expected_value=exp(-3.0)
-  )
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqDevTools&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# but apply it only to u[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Int&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_convergence&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
 Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function for the other aforementioned solvers, we obtain the convergence plot:&lt;/p&gt;
 
 
 
 
 
-  
-
-
-
-
-
 
 
 
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/weak_conv.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/weak_conv.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;weak_conv.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Note that the &lt;code&gt;SimplifiedEM&lt;/code&gt; and the &lt;code&gt;EM&lt;/code&gt; scheme fall on top of each other.
 &lt;code&gt;DRI1()&lt;/code&gt; achieves the smallest errors for a fixed &lt;code&gt;dt&lt;/code&gt; in this study.&lt;/p&gt;
@@ -2419,39 +2157,33 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 &lt;p&gt;Ultimately, we are not only interested in the general convergence slope of an algorithm but also in its speed.
 We&amp;rsquo;d like to select the fastest method depending on the permitted tolerance.
 This is commonly studied using work-precision diagram.
-Thanks to 
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;reltols = 1.0 ./ 4.0 .^ (1:4)
-abstols = reltols#[0.0 for i in eachindex(reltols)]
-setups = [
-          Dict(:alg=&amp;gt;DRI1(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;PL1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;EM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;SimplifiedEM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI2WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false)
-          ]
-test_dt = 1//10000
-appxsol_setup = Dict(:alg=&amp;gt;EM(), :dt=&amp;gt;test_dt)
-wp = @time WorkPrecisionSet(ensemble_prob,
-                            abstols,reltols,setups,test_dt;
-                            maxiters = 1e7,verbose=false,
-                            save_everystep=false,save_start=false,
-                            appxsol_setup = appxsol_setup,
-                            expected_value=exp(-3.0),
-                            trajectories=numtraj, error_estimate=:weak_final)
-plt = plot(wp;legend=:bottomleft)
-&lt;/code&gt;&lt;/pre&gt;
+Thanks to &lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;./&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;4.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;c&#34;&gt;#[0.0 for i in eachindex(reltols)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;PL1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;SimplifiedEM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI2WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10000&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WorkPrecisionSet&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;maxiters&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;verbose&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;error_estimate&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:weak_final&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
-
-
-
-
 
 
 
@@ -2459,88 +2191,83 @@ plt = plot(wp;legend=:bottomleft)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;WorkPrecision.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Therefore, &lt;code&gt;DRI1&lt;/code&gt; has the best performance in this non-commutative noise case if the error is supposed to stay below 1e-3.
 For larger permitted errors, the &lt;code&gt;SimplifiedEM&lt;/code&gt; scheme might be a good choice. However, the first order methods
 are outclassed when high precision is more of a concern.
-We plan to perform more in-depth benchmarks in the near future what will be reported on the 
-&lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
+We plan to perform more in-depth benchmarks in the near future what will be reported on the &lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
 &lt;h3 id=&#34;adaptive-step-size-control&#34;&gt;Adaptive step-size control&lt;/h3&gt;
-&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
+&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
 &lt;p&gt;To use adaptive step-size control, it is sufficient to set &lt;code&gt;adaptive=true&lt;/code&gt; (default setting). Optionally, one may also pass absolute and relative tolerances.&lt;/p&gt;
 &lt;p&gt;The following julia code simulates the stochastic version of the Brusselator equations with intitial condition&lt;/p&gt;
 &lt;p&gt;$$ X(t=0)=  \begin{pmatrix} 0.1 \\ 0\end{pmatrix},$$&lt;/p&gt;
 &lt;p&gt;on a time span $\mathbb{I}=[0, 100]$ for adaptive (&lt;code&gt;sol&lt;/code&gt;) and fixed step sizes (&lt;code&gt;sol_na&lt;/code&gt;):&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq, DiffEqNoiseProcess, Random
-using Plots
-using DiffEqGPU
-
-function prob_func(prob, i, repeat)
-  Random.seed!(seeds[i])
-  W = WienerProcess(0.0,0.0,0.0)
-  remake(prob,noise=W)
-end
-
-function brusselator_f!(du,u,p,t)
-  @inbounds begin
-    du[1] = (p[1]-1)*u[1]+p[1]*u[1]^2+(u[1]+1)^2*u[2]
-    du[2] = -p[1]*u[1]-p[1]*u[1]^2-(u[1]+1)^2*u[2]
-  end
-  nothing
-end
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqGPU&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# fix seeds&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.9f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleSummary&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fillalpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;time t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;X(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
-function scalar_noise!(du,u,p,t)
-  @inbounds begin
-    du[1] = p[2]*u[1]*(1+u[1])
-    du[2] = -p[2]*u[1]*(1+u[1])
-   end
-   nothing
-end
 
 
-# fix seeds
-seed = 100
-Random.seed!(seed)
-numtraj= 100
-seeds = rand(UInt, numtraj)
-W = WienerProcess(0.0,0.0,0.0)
 
 
-u0 = [-0.1f0,0.0f0]
-tspan = (0.0f0,100.0f0)
-p = [1.9f0,0.1f0]
 
-prob = SDEProblem(brusselator_f!,scalar_noise!,u0,tspan,p,noise=W)
 
-ensembleprob = EnsembleProblem(prob, prob_func = prob_func)
-
-sol = @time solve(ensembleprob,DRI1(),dt=0.1,EnsembleCPUArray(),trajectories=numtraj)
-sol_na = @time solve(ensembleprob,DRI1(),dt=0.8,adaptive=false,EnsembleCPUArray(),trajectories=numtraj)
-
-
-summ = EnsembleSummary(sol,0.0f0:0.5f0:100f0)
-pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quot;X(t)&amp;quot;, label= [&amp;quot;x₁(t)&amp;quot; &amp;quot;x₂(t)&amp;quot;], legend=true)
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
-
-
-
-
-
-  
 
 
 
@@ -2552,69 +2279,53 @@ pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quo
 
 
 
-&lt;figure &gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator_many_trajectories.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
+&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref3:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
 as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DifferentialEquations.EnsembleAnalysis
-meansol = timeseries_steps_mean(sol)
-meansol_na = timeseries_point_mean(sol_na,meansol.t)
-dts = []
-tmp1 = tspan[1]
-for tmp2 in meansol.t
-  global tmp1
-  push!(dts,tmp2-tmp1)
-  tmp1 = tmp2
-end
-#
-
-list_plots = []
-for i in 1:length(meansol.u)
-  l = @layout [a  b]
-  plt1 = plot(meansol[1, 1:i],meansol[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;adaptive&amp;quot;,
-         lw=2,
-         linecolor=1
-         )
-  plot!(meansol_na[1, 1:i],meansol_na[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;fixed step size&amp;quot;,
-         lw=2,
-         linecolor=2
-         )
-  pl2 = scatter(dts[1:i], xlabel = &amp;quot;step&amp;quot;, yaxis= &amp;quot;dtᵢ&amp;quot;, xlim = (0, length(meansol.u)),  ylim = (0.0, 2.3), legend=false)
-  plt = plot(plt1, pl2, layout = l)
-  push!(list_plots, plt)
-end
-
-anim = animate(list_plots,lw=2,every=1)
-
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleAnalysis&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_steps_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_point_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;global&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;#&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@layout&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;adaptive&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;fixed step size&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scatter&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;step&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;dtᵢ&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)),&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;2.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -2625,53 +2336,49 @@ anim = animate(list_plots,lw=2,every=1)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gpu-usage&#34;&gt;GPU usage&lt;/h3&gt;
 &lt;p&gt;All necessary tools to accelerate the simulation of (stochastic) differential equations on GPUs within the SciML
-ecosystem are collected in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
+ecosystem are collected in the &lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;Currently, bounds checking and return values are not allowed, i.e., functions must be
 written in the form:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f!(du,u,p,t)
-  @inbounds begin
-    du[1] = ..
-  end
-  nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;..&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
 Note that for some high weak order solvers GPU usage is not recommended as scalar indexing is used.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -2680,21 +2387,14 @@ Note that for some high weak order solvers GPU usage is not recommended as scala
       <link>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</link>
       <pubDate>Sat, 30 May 2020 15:10:33 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</guid>
-      <description>&lt;p&gt;First and foremost, I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
+      <description>&lt;p&gt;First and foremost, I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
 Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both  &lt;strong&gt;new high weak order solvers&lt;/strong&gt; as well as &lt;strong&gt;adjoint sensitivity methods&lt;/strong&gt; for stochastic differential equations (SDEs).
-The project is embedded within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
-Ultimately, the planned contributions will allow researchers to simulate (or even 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
+The project is embedded within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
+Ultimately, the planned contributions will allow researchers to simulate (or even &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
 Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.&lt;/p&gt;
 &lt;p&gt;This post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.&lt;/p&gt;
 &lt;h2 id=&#34;high-weak-order-solvers&#34;&gt;High Weak Order Solvers&lt;/h2&gt;
-&lt;p&gt;Currently, the 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
+&lt;p&gt;Currently, the &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
 In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.&lt;/p&gt;
 &lt;p&gt;However in many situations, we are only aiming for computing an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt;.
 In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods.
@@ -2702,7 +2402,7 @@ In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&
 As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.&lt;/p&gt;
 &lt;h3 id=&#34;second-order-runge-kutta-methods-for-ito-sdes&#34;&gt;Second order Runge-Kutta methods for Ito SDEs&lt;/h3&gt;
 &lt;p&gt;In the beginning of the community bonding period I finished the implementations of the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and &lt;code&gt;RI1()&lt;/code&gt;&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;. Specifically, only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn (in contrast to &lt;code&gt;m(m+1)/2&lt;/code&gt; from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.&lt;/p&gt;
-&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -2711,47 +2411,38 @@ $$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e7&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;. The code for a single  &lt;code&gt;dt&lt;/code&gt; reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-numtraj = 1e7
-u₀ = [1.0,1.0]
-function f!(du,u,p,t)
-  du[1] = -273//512*u[1]
-  du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-end
-function g!(du,u,p,t)
-  du[1,1] = 1//4*u[1]
-  du[1,2] = 1//16*u[1]
-  du[2,1] = (1-2*sqrt(2))/4*u[1]
-  du[2,2] = 1//10*u[1]+1//16*u[2]
-end
-dt = 1//8
-tspan = (0.0,10.0)
-prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-h(z) = z^2
-
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h(sol[end][1]),false)
-        )
-sol = solve(ensemble_prob, DRI1();
-        dt=dt,
-        save_start=false,
-        save_everystep=false,
-        weak_timeseries_errors=false,
-        weak_dense_errors=false,
-        trajectories=numtraj)
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
-&lt;pre&gt;&lt;code&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
-
-
-
-
-
-  
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;();&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
 
 
 
@@ -2763,29 +2454,26 @@ sol = solve(ensemble_prob, DRI1();
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/DRI1.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/DRI1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;DRI1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In the next couple of weeks, my focus will be on&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;adding other high weak order solvers,&lt;/li&gt;
 &lt;li&gt;implementing adaptive time stepping.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;More of our near-term goals are collected in this 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;More of our near-term goals are collected in this &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;adjoint-sensitivity-methods-for-sdes&#34;&gt;Adjoint Sensitivity Methods for SDEs&lt;/h2&gt;
-&lt;p&gt;In 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
+&lt;p&gt;In &lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
 &lt;p&gt;$$
 L = |f(p)-y|^2.
 $$&lt;/p&gt;
@@ -2794,50 +2482,46 @@ One possibility to obtain the gradient information for (stochastic) differential
 While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.&lt;/p&gt;
 &lt;p&gt;Alternatively to the &amp;ldquo;direct&amp;rdquo; AD approaches, the &lt;strong&gt;adjoint sensitivity method&lt;/strong&gt; can be used&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.
 Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).&lt;/p&gt;
-&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
+&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
 &lt;h3 id=&#34;reverse-sde-time-evolution&#34;&gt;Reverse SDE time evolution&lt;/h3&gt;
 &lt;p&gt;The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms.
 As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed.
 In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values.
-After some fixes for the 
-&lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
+After some fixes for the &lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
 &lt;p&gt;As an example, the code below computes first the forward evolution of an SDE&lt;/p&gt;
 &lt;p&gt;$$ dX  =  \alpha X dt + \beta X dW$$&lt;/p&gt;
 &lt;p&gt;with $\alpha=1.01$, $\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;  using StochasticDiffEq, DiffEqNoiseProcess
-
-  α=1.01
-  β=0.87
-
-  dt = 1e-3
-  tspan = (0.0,1.0)
-  u₀=1/2
-
-  tarray =  collect(tspan[1]:dt:tspan[2])
-
-  f!(du,u,p,t) = du .= α*u
-  g!(du,u,p,t) = du .= β*u
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# to make sure the plot is correct&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NoiseGrid&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  prob = SDEProblem(f!,g!,[u₀],tspan)
-  sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false)
 
-  _sol = deepcopy(sol) # to make sure the plot is correct
-  W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W))
-  prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1)
-  sol1 = solve(prob1,EulerHeun(),dt=dt)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
-
-
-
-
 
 
 
@@ -2845,18 +2529,12 @@ After some fixes for the
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/animation.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/animation.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;animation.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gradients-of-diagonal-sdes&#34;&gt;Gradients of diagonal SDEs&lt;/h3&gt;
 &lt;p&gt;I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.&lt;/p&gt;
@@ -2870,82 +2548,80 @@ $$&lt;/p&gt;
 &lt;p&gt;acting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\&lt;br&gt;
+\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\
 \frac{d L}{d \beta}  &amp;amp;= 2 \sum_i W(t_i) |X(t_i)|^2
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Test, LinearAlgebra
-using DiffEqSensitivity, StochasticDiffEq
-using Random
-
-seed = 100
-Random.seed!(seed)
-
-u₀ = [0.5]
-tstart = 0.0
-tend = 0.1
-dt = 0.005
-trange = (tstart, tend)
-t = tstart:dt:tend
-tarray = collect(t)
-
-function g(u,p,t)
-  sum(u.^2.0)
-end
-
-function dg!(out,u,p,t,i)
-  (out.=-2.0*u)
-end
-
-p2 = [1.01,0.87]
-
-f(u,p,t) = p[1]*u
-σ(u,p,t) = p[2]*u
-
-
-Random.seed!(seed)
-prob = SDEProblem(f,σ,u₀,trange,p2)
-sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true)
-res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint())
-
-
-noise = vec((@. sol.W(tarray)))
-Wextracted = [W[1][1] for W in noise]
-resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp = [resp1, resp2]
-
-@test isapprox(res_p&#39;, resp, rtol = 1e-6)
-# True
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.005&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:tend&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.=-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RKMil&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;interpretation&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:Stratonovich&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res_u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adjoint_sensitivities&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vec&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;isapprox&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# True&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;finish the current backsolve adjoint version,&lt;/li&gt;
 &lt;li&gt;allow for computing the gradients of non-commuting SDEs,&lt;/li&gt;
 &lt;li&gt;implement also an interpolation adjoint version,&lt;/li&gt;
 &lt;li&gt;benchmark it with respect to AD approaches&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -2981,6 +2657,183 @@ resp = [resp1, resp2]
       <description></description>
     </item>
     
+    <item>
+      <title>An example preprint / working paper</title>
+      <link>https://frankschae.github.io/publication/preprint/</link>
+      <pubDate>Sun, 07 Apr 2019 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/preprint/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
+</description>
+    </item>
+    
+    <item>
+      <title>Slides</title>
+      <link>https://frankschae.github.io/slides/example/</link>
+      <pubDate>Tue, 05 Feb 2019 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/slides/example/</guid>
+      <description>&lt;h1 id=&#34;create-slides-in-markdown-with-hugo-blox-builder&#34;&gt;Create slides in Markdown with Hugo Blox Builder&lt;/h1&gt;
+&lt;p&gt;&lt;a href=&#34;https://hugoblox.com/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Hugo Blox Builder&lt;/a&gt; | &lt;a href=&#34;https://docs.hugoblox.com/content/slides/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Documentation&lt;/a&gt;&lt;/p&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;features&#34;&gt;Features&lt;/h2&gt;
+&lt;ul&gt;
+&lt;li&gt;Efficiently write slides in Markdown&lt;/li&gt;
+&lt;li&gt;3-in-1: Create, Present, and Publish your slides&lt;/li&gt;
+&lt;li&gt;Supports speaker notes&lt;/li&gt;
+&lt;li&gt;Mobile friendly slides&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;controls&#34;&gt;Controls&lt;/h2&gt;
+&lt;ul&gt;
+&lt;li&gt;Next: &lt;code&gt;Right Arrow&lt;/code&gt; or &lt;code&gt;Space&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Previous: &lt;code&gt;Left Arrow&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Start: &lt;code&gt;Home&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Finish: &lt;code&gt;End&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Overview: &lt;code&gt;Esc&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Speaker notes: &lt;code&gt;S&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Fullscreen: &lt;code&gt;F&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Zoom: &lt;code&gt;Alt + Click&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://revealjs.com/pdf-export/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PDF Export&lt;/a&gt;&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;code-highlighting&#34;&gt;Code Highlighting&lt;/h2&gt;
+&lt;p&gt;Inline code: &lt;code&gt;variable&lt;/code&gt;&lt;/p&gt;
+&lt;p&gt;Code block:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;porridge&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s2&#34;&gt;&amp;#34;blueberry&amp;#34;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;porridge&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;s2&#34;&gt;&amp;#34;blueberry&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;nb&#34;&gt;print&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;Eating...&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;hr&gt;
+&lt;h2 id=&#34;math&#34;&gt;Math&lt;/h2&gt;
+&lt;p&gt;In-line math: $x + y = z$&lt;/p&gt;
+&lt;p&gt;Block math:&lt;/p&gt;
+&lt;p&gt;$$
+f\left( x \right) = ;\frac{{2\left( {x + 4} \right)\left( {x - 4} \right)}}{{\left( {x + 4} \right)\left( {x + 1} \right)}}
+$$&lt;/p&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;fragments&#34;&gt;Fragments&lt;/h2&gt;
+&lt;p&gt;Make content appear incrementally&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{% fragment %}} One {{% /fragment %}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{% fragment %}} **Two** {{% /fragment %}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{% fragment %}} Three {{% /fragment %}}
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Press &lt;code&gt;Space&lt;/code&gt; to play!&lt;/p&gt;
+&lt;span class=&#34;fragment &#34; &gt;
+  One
+&lt;/span&gt;
+&lt;span class=&#34;fragment &#34; &gt;
+  &lt;strong&gt;Two&lt;/strong&gt;
+&lt;/span&gt;
+&lt;span class=&#34;fragment &#34; &gt;
+  Three
+&lt;/span&gt;
+&lt;hr&gt;
+&lt;p&gt;A fragment can accept two optional parameters:&lt;/p&gt;
+&lt;ul&gt;
+&lt;li&gt;&lt;code&gt;class&lt;/code&gt;: use a custom style (requires definition in custom CSS)&lt;/li&gt;
+&lt;li&gt;&lt;code&gt;weight&lt;/code&gt;: sets the order in which a fragment appears&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;speaker-notes&#34;&gt;Speaker Notes&lt;/h2&gt;
+&lt;p&gt;Add speaker notes to your presentation&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-markdown&#34; data-lang=&#34;markdown&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{% speaker_note %}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;-&lt;/span&gt; Only the speaker can read these notes
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;-&lt;/span&gt; Press &lt;span class=&#34;sb&#34;&gt;`S`&lt;/span&gt; key to view
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  {{% /speaker_note %}}
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Press the &lt;code&gt;S&lt;/code&gt; key to view the speaker notes!&lt;/p&gt;
+&lt;aside class=&#34;notes&#34;&gt;
+  &lt;ul&gt;
+&lt;li&gt;Only the speaker can read these notes&lt;/li&gt;
+&lt;li&gt;Press &lt;code&gt;S&lt;/code&gt; key to view&lt;/li&gt;
+&lt;/ul&gt;
+
+&lt;/aside&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;themes&#34;&gt;Themes&lt;/h2&gt;
+&lt;ul&gt;
+&lt;li&gt;black: Black background, white text, blue links (default)&lt;/li&gt;
+&lt;li&gt;white: White background, black text, blue links&lt;/li&gt;
+&lt;li&gt;league: Gray background, white text, blue links&lt;/li&gt;
+&lt;li&gt;beige: Beige background, dark text, brown links&lt;/li&gt;
+&lt;li&gt;sky: Blue background, thin dark text, blue links&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+&lt;ul&gt;
+&lt;li&gt;night: Black background, thick white text, orange links&lt;/li&gt;
+&lt;li&gt;serif: Cappuccino background, gray text, brown links&lt;/li&gt;
+&lt;li&gt;simple: White background, black text, blue links&lt;/li&gt;
+&lt;li&gt;solarized: Cream-colored background, dark green text, blue links&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+
+&lt;section data-noprocess data-shortcode-slide
+  
+      
+      data-background-image=&#34;/media/boards.jpg&#34;
+  &gt;
+
+&lt;h2 id=&#34;custom-slide&#34;&gt;Custom Slide&lt;/h2&gt;
+&lt;p&gt;Customize the slide style and background&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-markdown&#34; data-lang=&#34;markdown&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;slide&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;background-image&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;/media/boards.jpg&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;slide&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;background-color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;#0000FF&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;slide&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;class&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;my-style&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;hr&gt;
+&lt;h2 id=&#34;custom-css-example&#34;&gt;Custom CSS Example&lt;/h2&gt;
+&lt;p&gt;Let&amp;rsquo;s make headers navy colored.&lt;/p&gt;
+&lt;p&gt;Create &lt;code&gt;assets/css/reveal_custom.css&lt;/code&gt; with:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-css&#34; data-lang=&#34;css&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nc&#34;&gt;reveal&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;section&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;h1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nc&#34;&gt;reveal&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;section&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nc&#34;&gt;reveal&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;section&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;h3&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;kc&#34;&gt;navy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;hr&gt;
+&lt;h1 id=&#34;questions&#34;&gt;Questions?&lt;/h1&gt;
+&lt;p&gt;&lt;a href=&#34;https://discord.gg/z8wNYzb&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Ask&lt;/a&gt;&lt;/p&gt;
+&lt;p&gt;&lt;a href=&#34;https://docs.hugoblox.com/content/slides/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Documentation&lt;/a&gt;&lt;/p&gt;
+</description>
+    </item>
+    
+    <item>
+      <title>An example journal article</title>
+      <link>https://frankschae.github.io/publication/journal-article/</link>
+      <pubDate>Tue, 01 Sep 2015 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/journal-article/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Click the &lt;em&gt;Cite&lt;/em&gt; button above to demo the feature to enable visitors to import publication metadata into their reference management software.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
+</description>
+    </item>
+    
+    <item>
+      <title>An example conference paper</title>
+      <link>https://frankschae.github.io/publication/conference-paper/</link>
+      <pubDate>Mon, 01 Jul 2013 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/conference-paper/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Click the &lt;em&gt;Cite&lt;/em&gt; button above to demo the feature to enable visitors to import publication metadata into their reference management software.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
+</description>
+    </item>
+    
     <item>
       <title>Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers</title>
       <link>https://frankschae.github.io/project/scattering/</link>
diff --git a/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js b/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js
new file mode 100644
index 0000000..7091c9e
--- /dev/null
+++ b/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js
@@ -0,0 +1,8 @@
+/*! jQuery v3.6.4 | (c) OpenJS Foundation and other contributors | jquery.org/license *//*!
+  * Bootstrap v4.6.1 (https://getbootstrap.com/)
+  * Copyright 2011-2021 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)
+  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+  *//*! instant.page v5.2.0 - (C) 2019-2023 Alexandre Dieulot - https://instant.page/license */!function(e,t){"use strict";"object"==typeof module&&"object"==typeof module.exports?module.exports=e.document?t(e,!0):function(e){if(!e.document)throw new Error("jQuery requires a window with a document");return t(e)}:t(e)}("undefined"!=typeof window?window:this,function(e,t){"use strict";var n,l,m,j,M,F,T,N,R,H,W,X,Z,g=[],Fe=Object.getPrototypeOf,b=g.slice,ze=g.flat?function(e){return g.flat.call(e)}:function(e){return g.concat.apply([],e)},_e=g.push,Q=g.indexOf,J={},Be=J.toString,te=J.hasOwnProperty,Ge=te.toString,Ht=Ge.call(Object),a={},o=function(e){return"function"==typeof e&&"number"!=typeof e.nodeType&&"function"!=typeof e.item},k=function(e){return e!=null&&e===e.window},i=e.document,Rt={type:!0,src:!0,nonce:!0,noModule:!0},oe,se,_t,yt,bt,vt,ut,ce,Le,ye,ot,it,at,ht,ft,pt,jt,ge,ne,Ye,He,we,Me,Ae,ke;function lt(e,t,n){var s,a,o=(n=n||i).createElement("script");if(o.text=e,t)for(s in 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diff --git a/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js b/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js
new file mode 100644
index 0000000..e1933b1
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+/*! Hugo Blox Builder v5.9.7 | https://hugoblox.com/ */
+/*! Copyright 2016-present George Cushen (https://georgecushen.com/) */
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new file mode 100644
index 0000000..7b627ec
--- /dev/null
+++ b/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js
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+function m(){if($("#map").length){let n=$("#map-provider").val(),e=$("#map-lat").val(),t=$("#map-lng").val(),o=parseInt($("#map-zoom").val()),p=$("#map-dir").val(),l=$("#map-api-key").val();if(n==="google")new GMaps({div:"#map",lat:e,lng:t,zoom:o,zoomControl:!0,zoomControlOpt:{style:"SMALL",position:"TOP_LEFT"},streetViewControl:!1,mapTypeControl:!1,gestureHandling:"cooperative"}).addMarker({lat:e,lng:t,click:function(){let r="https://www.google.com/maps/place/"+encodeURIComponent(p)+"/@"+e+","+t+"/";window.open(r,"_blank")},title:p});else{let a=new L.map("map").setView([e,t],o);n==="mapbox"&&l.length?L.tileLayer("https://api.mapbox.com/styles/v1/{id}/tiles/{z}/{x}/{y}?access_token={accessToken}",{attribution:'Map data &copy; <a href="http://openstreetmap.org">OpenStreetMap</a> contributors, <a href="http://creativecommons.org/licenses/by-sa/2.0/">CC-BY-SA</a>, Imagery \xA9 <a href="http://mapbox.com">Mapbox</a>',tileSize:512,maxZoom:18,zoomOffset:-1,id:"mapbox/streets-v11",accessToken:l}).addTo(a):L.tileLayer("https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png",{maxZoom:19,attribution:'&copy; <a href="http://www.openstreetmap.org/copyright">OpenStreetMap</a>'}).addTo(a);let r=L.marker([e,t]).addTo(a),i=e+","+t+"#map="+o+"/"+e+"/"+t+"&layers=N";r.bindPopup(p+'<p><a href="https://www.openstreetmap.org/directions?engine=osrm_car&route='+i+'">Routing via OpenStreetMap</a></p>')}}}document.addEventListener("DOMContentLoaded",function(){m()});
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new file mode 100644
index 0000000..1673d20
--- /dev/null
+++ b/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js
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new file mode 100644
index 0000000..7fe1540
--- /dev/null
+++ b/js/wowchemy-slides.js
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+(() => {
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+
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+    });
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+  var keysToCamelCase = function(o) {
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+      });
+      return n;
+    } else if (isArray(o)) {
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+      });
+    }
+    return o;
+  };
+  var pluginOptions = {};
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+  pluginOptions = keysToCamelCase(pluginOptions);
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+  }
+  if (pluginOptions.menu_enabled) {
+    enabledPlugins.push(RevealMenu);
+  }
+  pluginOptions["plugins"] = enabledPlugins;
+  Reveal.initialize(pluginOptions);
+  if (typeof slides.diagram === "undefined") {
+    slides.diagram = false;
+  }
+  if (slides.diagram) {
+    mermaidOptions = {};
+    if (typeof slides.diagram_options !== "undefined") {
+      mermaidOptions = slides.diagram_options;
+    }
+    mermaidOptions["startOnLoad"] = false;
+    mermaid.initialize(mermaidOptions);
+    let renderMermaidDiagrams = function renderMermaidDiagrams2(event) {
+      let mermaidDivs = event.currentSlide.querySelectorAll(".mermaid:not(.done)");
+      let indices = Reveal.getIndices();
+      let pageno = `${indices.h}-${indices.v}`;
+      mermaidDivs.forEach(function(mermaidDiv, i) {
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+          mermaidDiv.innerHTML = svgCode;
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+        };
+        let graphDefinition = mermaidDiv.textContent;
+        mermaid.mermaidAPI.render(`mermaid${pageno}-${i}`, graphDefinition, insertSvg);
+      });
+      Reveal.layout();
+    };
+    Reveal.on("ready", (event) => renderMermaidDiagrams(event));
+    Reveal.on("slidechanged", (event) => renderMermaidDiagrams(event));
+  }
+  var mermaidOptions;
+})();
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+  "theme_color": "#1565c0",
+  "background_color": "#1565c0",
+  "icons": [{
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+    "sizes": "192x192",
+    "type": "image/png"
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+  "display": "standalone",
+  "start_url": "/?utm_source=web_app_manifest"
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-      <p>
-<a href="https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best" target="_blank" rel="noopener">Differentiable programming (∂P)</a>, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup>. The Julia<sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> language is 
-<a href="https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md" target="_blank" rel="noopener">well suited for ∂P</a>, see also Chris' article<sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup> for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance <em>and</em> accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>, for example, by focusing on learning only the parts of the model that are actually missing<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> <sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>. In the context of quantum control, we have demonstrated the power of this framework for closed<sup id="fnref:6"><a href="#fn:6" class="footnote-ref" role="doc-noteref">6</a></sup> and 
-<a href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=3&amp;t=12s" target="_blank" rel="noopener">open quantum systems</a><sup id="fnref:7"><a href="#fn:7" class="footnote-ref" role="doc-noteref">7</a></sup>.</p>
-<p>∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, 
-<a href="https://juliadiff.org/" target="_blank" rel="noopener">a large variety of software implementations</a> with different 
-<a href="https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083" target="_blank" rel="noopener">pros and cons</a> exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by 
-<a href="https://github.com/JuliaDiff/ChainRules.jl" target="_blank" rel="noopener">ChainRules.jl</a> empowers this idea of a 
-<a href="http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/" target="_blank" rel="noopener">Glue AD</a> where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.</p>
-<p>
-<a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a> has started to 
-<a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1" target="_blank" rel="noopener">implement a high level API for differentiation</a> that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, 
-<a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl" target="_blank" rel="noopener">AbstractDifferentiation.jl</a>, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl" target="_blank" rel="noopener">DiffEqSensitivity.jl</a> within the 
-<a href="https://sciml.ai/" target="_blank" rel="noopener">SciML</a> software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&rsquo;ve started to fix remaining errors of the 
-<a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1" target="_blank" rel="noopener">initial PR</a>.</p>
-<p>The interested reader is encouraged to look at Mohamed&rsquo;s 
-<a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1" target="_blank" rel="noopener">first PR</a> for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.</p>
+      <p><a href="https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best" target="_blank" rel="noopener">Differentiable programming (∂P)</a>, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup>. The Julia<sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> language is <a href="https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md" target="_blank" rel="noopener">well suited for ∂P</a>, see also Chris&rsquo; article<sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup> for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance <em>and</em> accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>, for example, by focusing on learning only the parts of the model that are actually missing<sup id="fnref1:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> <sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>. In the context of quantum control, we have demonstrated the power of this framework for closed<sup id="fnref:6"><a href="#fn:6" class="footnote-ref" role="doc-noteref">6</a></sup> and <a href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=3&amp;t=12s" target="_blank" rel="noopener">open quantum systems</a><sup id="fnref:7"><a href="#fn:7" class="footnote-ref" role="doc-noteref">7</a></sup>.</p>
+<p>∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, <a href="https://juliadiff.org/" target="_blank" rel="noopener">a large variety of software implementations</a> with different <a href="https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083" target="_blank" rel="noopener">pros and cons</a> exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by <a href="https://github.com/JuliaDiff/ChainRules.jl" target="_blank" rel="noopener">ChainRules.jl</a> empowers this idea of a <a href="http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/" target="_blank" rel="noopener">Glue AD</a> where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.</p>
+<p><a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a> has started to <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1" target="_blank" rel="noopener">implement a high level API for differentiation</a> that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl" target="_blank" rel="noopener">AbstractDifferentiation.jl</a>, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that <a href="https://github.com/SciML/DiffEqSensitivity.jl" target="_blank" rel="noopener">DiffEqSensitivity.jl</a> within the <a href="https://sciml.ai/" target="_blank" rel="noopener">SciML</a> software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&rsquo;ve started to fix remaining errors of the <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1" target="_blank" rel="noopener">initial PR</a>.</p>
+<p>The interested reader is encouraged to look at Mohamed&rsquo;s <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1" target="_blank" rel="noopener">first PR</a> for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.</p>
 <h2 id="optimization-of-the-rosenbrock-function">Optimization of the Rosenbrock function</h2>
-<p>The 
-<a href="https://en.wikipedia.org/wiki/Rosenbrock_function" target="_blank" rel="noopener">Rosenbrock function</a> is defined by</p>
+<p>The <a href="https://en.wikipedia.org/wiki/Rosenbrock_function" target="_blank" rel="noopener">Rosenbrock function</a> is defined by</p>
 <p>$$
 g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2.
 $$</p>
 <p>The function $g$ has a global minimum at $(x_1^\star, x_2^\star)= (a, a^2)$ with $g(x_1^\star, x_2^\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.</p>
-<p>Let us now implement the 
-<a href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm" target="_blank" rel="noopener">Gauss–Newton algorithm</a> to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$</p>
+<p>Let us now implement the <a href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm" target="_blank" rel="noopener">Gauss–Newton algorithm</a> to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$</p>
 <p>$$
 S({\bf x}) = \frac{1}{2} \sum_{i=1}^M f_i({\bf x})^2.
 $$</p>
@@ -686,298 +869,275 @@ <h2 id="optimization-of-the-rosenbrock-function">Optimization of the Rosenbrock
 <p>$$
 {\bf x}^{k+1} = {\bf x}^k - \alpha_k \left(J^T J \right)^{-1} J^T f({\bf x}^k),
 $$
-where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a 
-<a href="https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel" target="_blank" rel="noopener">line search subroutine</a>.</p>
+where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a <a href="https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel" target="_blank" rel="noopener">line search subroutine</a>.</p>
 <p>The following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\bf x^\star}=(1,1)$ and our initial guess ${\bf x_0}=(0,-0.1)$.</p>
-<pre><code class="language-julia">using Pkg
-path = @__DIR__
-cd(path); Pkg.activate(&quot;.&quot;); Pkg.instantiate()
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Pkg</span>
+</span></span><span class="line"><span class="cl"><span class="n">path</span> <span class="o">=</span> <span class="nd">@__DIR__</span>
+</span></span><span class="line"><span class="cl"><span class="n">cd</span><span class="p">(</span><span class="n">path</span><span class="p">);</span> <span class="n">Pkg</span><span class="o">.</span><span class="n">activate</span><span class="p">(</span><span class="s">&#34;.&#34;</span><span class="p">);</span> <span class="n">Pkg</span><span class="o">.</span><span class="n">instantiate</span><span class="p">()</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c">## AbstractDifferentiation is not released yet!!</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">AbstractDifferentiation</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Test</span><span class="p">,</span> <span class="n">LinearAlgebra</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">FiniteDifferences</span><span class="p">,</span> <span class="n">ForwardDiff</span><span class="p">,</span> <span class="n">Zygote</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Enzyme</span><span class="p">,</span> <span class="n">UnPack</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Plots</span><span class="p">,</span> <span class="n">LaTeXStrings</span>
+</span></span><span class="line"><span class="cl"><span class="c"># using Diffractor: ∂⃖¹ ## Diffractor needs &gt;julia@1.6</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c">## Rosenbrock function</span>
+</span></span><span class="line"><span class="cl"><span class="c"># R: R^2 -&gt; R: x -&gt; (a-x₁)² + b(x₂-x₁²)²</span>
+</span></span><span class="line"><span class="cl"><span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">^</span><span class="mi">2</span> <span class="o">+</span> <span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">^</span><span class="mi">2</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># visualization</span>
+</span></span><span class="line"><span class="cl"><span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">100.0</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x₀</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">,</span><span class="o">-</span><span class="mf">0.1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">xopt</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">1.0</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">do_plot</span> <span class="o">=</span> <span class="nb">true</span>
+</span></span><span class="line"><span class="cl"><span class="k">if</span> <span class="n">do_plot</span>    
+</span></span><span class="line"><span class="cl">    <span class="n">x₁</span><span class="p">,</span> <span class="n">x₂</span> <span class="o">=</span> <span class="o">-</span><span class="mf">2.0</span><span class="o">:</span><span class="mf">0.01</span><span class="o">:</span><span class="mf">2.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.6</span><span class="o">:</span><span class="mf">0.01</span><span class="o">:</span><span class="mf">3.5</span>
+</span></span><span class="line"><span class="cl">    <span class="n">z</span> <span class="o">=</span> <span class="n">Surface</span><span class="p">((</span><span class="n">x₁</span><span class="p">,</span><span class="n">x₂</span><span class="p">)</span><span class="o">-&gt;</span><span class="n">g</span><span class="p">([</span><span class="n">x₁</span><span class="p">,</span><span class="n">x₂</span><span class="p">],</span><span class="n">p</span><span class="p">),</span> <span class="n">x₁</span><span class="p">,</span> <span class="n">x₂</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">pl1</span> <span class="o">=</span> <span class="n">surface</span><span class="p">(</span><span class="n">x₁</span><span class="p">,</span><span class="n">x₂</span><span class="p">,</span><span class="n">z</span><span class="p">,</span> <span class="n">linealpha</span> <span class="o">=</span> <span class="mf">0.3</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">cgrad</span><span class="p">(</span><span class="ss">:thermal</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="ss">:exp</span><span class="p">),</span> <span class="n">colorbar</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">                <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span><span class="n">camera</span> <span class="o">=</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">50</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">                <span class="n">xlabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;x_1&#34;</span><span class="p">,</span> <span class="n">ylabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;x_2&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">    <span class="n">pl2</span> <span class="o">=</span> <span class="n">heatmap</span><span class="p">(</span><span class="n">x₁</span><span class="p">,</span><span class="n">x₂</span><span class="p">,</span><span class="n">z</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">cgrad</span><span class="p">(</span><span class="ss">:thermal</span><span class="p">,</span> <span class="n">scale</span> <span class="o">=</span> <span class="ss">:exp</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">                <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">                <span class="n">xlabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;x_1&#34;</span><span class="p">,</span> <span class="n">ylabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;x_2&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">scatter!</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span> <span class="p">[(</span><span class="n">x₀</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">x₀</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span> <span class="n">label</span><span class="o">=</span><span class="sa">L</span><span class="s">&#34;x_0&#34;</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">markershape</span> <span class="o">=</span> <span class="ss">:circle</span><span class="p">,</span> <span class="n">markersize</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">markercolor</span> <span class="o">=</span> <span class="ss">:green</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">scatter!</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span> <span class="p">[(</span><span class="n">xopt</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">xopt</span><span class="p">[</span><span class="mi">2</span><span class="p">])],</span><span class="n">label</span><span class="o">=</span><span class="sa">L</span><span class="s">&#34;x^\star&#34;</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="n">markershape</span> <span class="o">=</span> <span class="ss">:star</span><span class="p">,</span> <span class="n">markersize</span> <span class="o">=</span> <span class="mi">10</span><span class="p">,</span> <span class="n">markercolor</span> <span class="o">=</span> <span class="ss">:red</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">    <span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span><span class="n">pl2</span><span class="p">,</span> <span class="n">layout</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">    <span class="n">savefig</span><span class="p">(</span><span class="n">pl</span><span class="p">,</span> <span class="s">&#34;Rosenbrock.png&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div>
 
-## AbstractDifferentiation is not released yet!!
 
-using AbstractDifferentiation
-using Test, LinearAlgebra
-using FiniteDifferences, ForwardDiff, Zygote
-using Enzyme, UnPack
-using Plots, LaTeXStrings
-# using Diffractor: ∂⃖¹ ## Diffractor needs &gt;julia@1.6
 
-## Rosenbrock function
-# R: R^2 -&gt; R: x -&gt; (a-x₁)² + b(x₂-x₁²)²
-g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2
 
-# visualization
-p = [1.0,100.0]
-x₀ = [0.0,-0.1]
-xopt = [1.0,1.0]
 
-do_plot = true
-if do_plot    
-    x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5
-    z = Surface((x₁,x₂)-&gt;g([x₁,x₂],p), x₁, x₂)
-    pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true,
-                labelfontsize=20,camera = (3,50),
-                xlabel = L&quot;x_1&quot;, ylabel = L&quot;x_2&quot;)
 
-    pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp),
-                labelfontsize=20,
-                xlabel = L&quot;x_1&quot;, ylabel = L&quot;x_2&quot;)
-    scatter!(pl2, [(x₀[1],x₀[2])], label=L&quot;x_0&quot;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green)
-    scatter!(pl2, [(xopt[1],xopt[2])],label=L&quot;x^\star&quot;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red)
 
-    pl = plot(pl1,pl2, layout=(2,1))
-    savefig(pl, &quot;Rosenbrock.png&quot;)
-end
-</code></pre>
 
 
 
 
 
-  
 
 
 
 
 
-
-
-
-
-
-
-<figure >
-
-
-  <a data-fancybox="" href="/img/Rosenbrock.png" >
-
-
-  <img src="/img/Rosenbrock.png" alt=""  >
-</a>
-
-
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="Rosenbrock.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>To apply the Gauss-Newton algorithm to the Rosenbrock function $g$, we first cast $g$ into an appropriate form fulfilling $S({\bf x})$, i.e., we use:</p>
 <p>$$
 f:\mathbb{R}^2\rightarrow\mathbb{R}^2:  {\bf x} \mapsto \begin{pmatrix}
-f_1({\bf x}) \\<br>
-f_2({\bf x}) \\<br>
+f_1({\bf x}) \\
+f_2({\bf x}) \\
 \end{pmatrix} = \begin{pmatrix}
-\sqrt{2}(a-x_1) \\<br>
-\sqrt{2b}(x_2-x_1^2)\\<br>
+\sqrt{2}(a-x_1) \\
+\sqrt{2b}(x_2-x_1^2)\\
 \end{pmatrix},
 $$</p>
 <p>instead of $g$. We can easily compute the Jacobian of $f$ manually</p>
 <p>$$
 J =  \begin{pmatrix}
--\sqrt{2} &amp; 0 \\<br>
--2x_1\sqrt{2b} &amp; \sqrt{2b} \\<br>
+-\sqrt{2} &amp; 0 \\
+-2x_1\sqrt{2b} &amp; \sqrt{2b} \\
 \end{pmatrix}.
 $$</p>
 <p>We can then implement a (simple, non-optimized) version of the Gauss-Newton algorithm as follows.</p>
-<pre><code class="language-julia"># bring Rosenbrock function into the form &quot;sum of squares of functions&quot;
-f1(x,p) = convert(eltype(x),sqrt(2))*(p[1]-x[1])
-f2(x,p) = convert(eltype(x),sqrt(2*p[2]))*(x[2]-x[1]^2)
-f(x,p) = [f1(x,p),f2(x,p)]
-function f(res,x,p) # Enzyme works with inplace functions
-	res[1] = f1(x,p)
-	res[2] = f2(x,p)
-	return nothing
-end
-
-## manually pre-defined Jacobian
-function Jacobian(x,p)
-  [-convert(eltype(x),sqrt(2))   0
-  -2*x[1]*convert(eltype(x),sqrt(2*p[2]))  convert(eltype(x),sqrt(2*p[2]))]
-end
-
-## Gauss-Newton scheme
-function GaussNewton!(xs, x, p; maxiter=8, backend=nothing)
-    for i=1:maxiter
-        x = step(x, p, backend)
-        @info i
-        @show x
-        push!(xs, x)
-    end
-    return xs, x
-end
-done(x,x2,p) = g(x2,p) &lt; g(x,p)
-function step(x, p, backend::Nothing, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = Jacobian(x,p)
-    d = -inv(J'*J)*J'*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-</code></pre>
-<p>When we run the algorithm, we find the global minimum after about the 7th iteration.</p>
-<pre><code class="language-julia">xs = [x₀]
-GaussNewton!(xs, x₀, p)
-</code></pre>
-<pre><code class="language-julia"># output:
-[ Info: 1 ]
-x = [0.125, -0.08750000000000001]
-[ Info: 2 ]
-x = [0.234375, -0.047265625000000006]
-[ Info: 3 ]
-x = [0.4257812499999995, 0.06800537109374968]
-[ Info: 4 ]
-x = [0.5693359374999986, 0.21857223510742047]
-[ Info: 5 ]
-x = [0.784667968749996, 0.5165503501892037]
-[ Info: 6 ]
-x = [0.9999999999999961, 0.9536321163177449]
-[ Info: 7 ]
-x = [0.9999999999999989, 0.9999999999999999]
-[ Info: 8 ]
-x = [1.0, 1.0]
-</code></pre>
-<p>If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of 
-<a href="https://github.com/JuliaDiff/FiniteDifferences.jl" target="_blank" rel="noopener">FiniteDifferences.jl</a> as a new primitive operation.</p>
-<pre><code class="language-julia">## FiniteDifferences
-struct FDMBackend{A} &lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend() = FDMBackend(central_fdm(5, 1))
-const fdm_backend = FDMBackend()
-# Minimal interface
-AD.@primitive function jacobian(ab::FDMBackend, f, xs...)
-    return FiniteDifferences.jacobian(ab.alg, f, xs...)
-end
-
-# AD Jacobian returns tuple
-# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]
-# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]
-
-@test AD.jacobian(fdm_backend, x-&gt;f(x,p), x₀)[1] ≈ Jacobian(x₀, p)
-@test AD.jacobian(fdm_backend, f, x₀, p)[1] ≈ Jacobian(x₀, p)
-</code></pre>
-<p>After overloading the <code>step</code> function, we can run the Gauss-Newton algorithm as follows:</p>
-<pre><code class="language-julia">function step(x, p, backend, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = AD.jacobian(backend, f, x, p)[1]
-    d = -inv(J'*J)*J'*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-
-
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=fdm_backend)
-</code></pre>
-<p>If we want to use reverse-mode AD instead, for example via 
-<a href="https://github.com/FluxML/Zygote.jl" target="_blank" rel="noopener">Zygote.jl</a>, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.</p>
-<pre><code class="language-julia">## Zygote
-struct ZygoteBackend &lt;: AD.AbstractReverseMode end
-const zygote_backend = ZygoteBackend()
-AD.@primitive function pullback_function(ab::ZygoteBackend, f, xs...)
-    return function (vs)
-        # Supports only single output
-        _, back = Zygote.pullback(f, xs...)
-        if vs isa AbstractVector
-            back(vs)
-        else
-            @assert length(vs) == 1
-            back(vs[1])
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(zygote_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=zygote_backend)
-</code></pre>
-<p>Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (
-<a href="https://github.com/JuliaDiff/ForwardDiff.jl" target="_blank" rel="noopener">ForwardDiff.jl</a>)</p>
-<pre><code class="language-julia">## ForwardDiff
-struct ForwardDiffBackend &lt;: AD.AbstractForwardMode end
-const forwarddiff_backend = ForwardDiffBackend()
-AD.@primitive function pushforward_function(ab::ForwardDiffBackend, f, xs...)
-    # jvp = f'(x)*v, i.e., differentiate f(x + h*v) wrt h at 0
-    return function (vs)
-        if xs isa Tuple
-            @assert length(xs) &lt;= 2
-            if length(xs) == 1
-                (ForwardDiff.derivative(h-&gt;f(xs[1]+h*vs[1]),0),)
-            else
-                ForwardDiff.derivative(h-&gt;f(xs[1]+h*vs[1], xs[2]+h*vs[2]),0)
-            end
-        else
-            ForwardDiff.derivative(h-&gt;f(xs+h*vs),0)
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(forwarddiff_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=forwarddiff_backend)
-</code></pre>
-<p>where we have used that the Jacobian-vector product $f'(x)v$, i.e., the primitives of forward-mode AD, can be computed by 
-<a href="https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12" target="_blank" rel="noopener">differentiating $f(x + hv)$ with respect to $h$ at 0</a>.</p>
-<p>Many AD packages, such as Zygote, have troubles with mutating functions. 
-<a href="https://github.com/wsmoses/Enzyme.jl" target="_blank" rel="noopener">Enzyme.jl</a> is one of the exceptions. Additionally, it is very fast and has further improved the performance of the 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944" target="_blank" rel="noopener">adjoints implemented within the DiffEqSensitivity package</a>.</p>
-<pre><code class="language-julia">## Enzyme
-struct EnzymeBackend{T1,T2,T3,T4} &lt;: AD.AbstractReverseMode
-    out::T1
-    λ::T2
-    ∂f_∂x::T3
-    ∂f_∂p::T4
-end
-
-out = zero(x₀)
-λ = zero(x₀)
-∂f_∂x = zero(x₀)
-∂f_∂p = zero(p)
-
-const enzyme_backend = EnzymeBackend(out,λ,∂f_∂x,∂f_∂p)
-AD.@primitive function pullback_function(ab::EnzymeBackend, f, xs...)
-    return function (vs)  
-        # enzyme works only with inplace functions
-        if !(vs isa AbstractVector)
-            @assert length(vs) == 1 # Supports only single output
-            vs = vs[1]
-        end
-
-        if xs isa Tuple
-            @assert length(xs) == 2  # hard-coded for use case with two inputs
-            x₀ = xs[1]
-            p = xs[2]
-        end
-
-        @unpack out, λ, ∂f_∂x, ∂f_∂p = ab # cached in the struct, could also be created in here
-
-        ∂f_∂x .*= false
-        ∂f_∂p .*= false
-        out .*= false
-
-        copyto!(λ, vs)
-
-        autodiff(Duplicated(out, λ), Duplicated(x₀, ∂f_∂x), Duplicated(p, ∂f_∂p)) do _out,_x, _p
-            f(_out,_x,_p)
-        end
-        return (∂f_∂x,∂f_∂p)
-    end
-end
-AD.isinplace(ab::EnzymeBackend) = true
-AD.primalvalue(ab::EnzymeBackend, nothing, f, xs) = (f(ab.out,xs...);return ab.out)
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(enzyme_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=enzyme_backend)
-</code></pre>
-<p>Note that we have declared the Enzyme backend as <code>inplace</code> (which is important for internal control flow) and specified a <code>primalvalue</code> function returning the primal value of the forward pass.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># bring Rosenbrock function into the form &#34;sum of squares of functions&#34;</span>
+</span></span><span class="line"><span class="cl"><span class="n">f1</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span> <span class="o">=</span> <span class="n">convert</span><span class="p">(</span><span class="n">eltype</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">*</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl"><span class="n">f2</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span> <span class="o">=</span> <span class="n">convert</span><span class="p">(</span><span class="n">eltype</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">^</span><span class="mi">2</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span> <span class="o">=</span> <span class="p">[</span><span class="n">f1</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">),</span><span class="n">f2</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)]</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span> <span class="c"># Enzyme works with inplace functions</span>
+</span></span><span class="line"><span class="cl">	<span class="n">res</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">f1</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">	<span class="n">res</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">f2</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">	<span class="k">return</span> <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c">## manually pre-defined Jacobian</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">Jacobian</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="p">[</span><span class="o">-</span><span class="n">convert</span><span class="p">(</span><span class="n">eltype</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>   <span class="mi">0</span>
+</span></span><span class="line"><span class="cl">  <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">convert</span><span class="p">(</span><span class="n">eltype</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>  <span class="n">convert</span><span class="p">(</span><span class="n">eltype</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]))]</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c">## Gauss-Newton scheme</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">GaussNewton!</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">;</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="nb">nothing</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">for</span> <span class="n">i</span><span class="o">=</span><span class="mi">1</span><span class="o">:</span><span class="n">maxiter</span>
+</span></span><span class="line"><span class="cl">        <span class="n">x</span> <span class="o">=</span> <span class="n">step</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">backend</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="nd">@info</span> <span class="n">i</span>
+</span></span><span class="line"><span class="cl">        <span class="nd">@show</span> <span class="n">x</span>
+</span></span><span class="line"><span class="cl">        <span class="n">push!</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">end</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="n">xs</span><span class="p">,</span> <span class="n">x</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="n">done</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">x2</span><span class="p">,</span><span class="n">p</span><span class="p">)</span> <span class="o">=</span> <span class="n">g</span><span class="p">(</span><span class="n">x2</span><span class="p">,</span><span class="n">p</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">step</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">backend</span><span class="o">::</span><span class="kt">Nothing</span><span class="p">,</span> <span class="n">α</span><span class="o">=</span><span class="mi">1</span><span class="o">//</span><span class="mi">1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">x2</span> <span class="o">=</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="k">while</span> <span class="o">!</span><span class="n">done</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">x2</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">J</span> <span class="o">=</span> <span class="n">Jacobian</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">d</span> <span class="o">=</span> <span class="o">-</span><span class="n">inv</span><span class="p">(</span><span class="n">J</span><span class="o">&#39;*</span><span class="n">J</span><span class="p">)</span><span class="o">*</span><span class="n">J</span><span class="o">&#39;*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">copyto!</span><span class="p">(</span><span class="n">x2</span><span class="p">,</span><span class="n">x</span> <span class="o">+</span> <span class="n">α</span><span class="o">*</span><span class="n">d</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">α</span> <span class="o">=</span> <span class="n">α</span><span class="o">//</span><span class="mi">2</span>
+</span></span><span class="line"><span class="cl">  <span class="k">end</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="n">x2</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><p>When we run the algorithm, we find the global minimum after about the 7th iteration.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="n">xs</span> <span class="o">=</span> <span class="p">[</span><span class="n">x₀</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">GaussNewton!</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span></code></pre></div><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># output:</span>
+</span></span><span class="line"><span class="cl"><span class="p">[</span> <span class="n">Info</span><span class="o">:</span> <span class="mi">1</span> <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.125</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.08750000000000001</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="p">[</span> <span class="n">Info</span><span class="o">:</span> <span class="mi">2</span> <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.234375</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.047265625000000006</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="p">[</span> <span class="n">Info</span><span class="o">:</span> <span class="mi">3</span> <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.4257812499999995</span><span class="p">,</span> <span class="mf">0.06800537109374968</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="p">[</span> <span class="n">Info</span><span class="o">:</span> <span class="mi">4</span> <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.5693359374999986</span><span class="p">,</span> <span class="mf">0.21857223510742047</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="p">[</span> <span class="n">Info</span><span class="o">:</span> <span class="mi">5</span> <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.784667968749996</span><span class="p">,</span> <span class="mf">0.5165503501892037</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="p">[</span> <span class="n">Info</span><span class="o">:</span> <span class="mi">6</span> <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.9999999999999961</span><span class="p">,</span> <span class="mf">0.9536321163177449</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="p">[</span> <span class="n">Info</span><span class="o">:</span> <span class="mi">7</span> <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.9999999999999989</span><span class="p">,</span> <span class="mf">0.9999999999999999</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="p">[</span> <span class="n">Info</span><span class="o">:</span> <span class="mi">8</span> <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">x</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">]</span>
+</span></span></code></pre></div><p>If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of <a href="https://github.com/JuliaDiff/FiniteDifferences.jl" target="_blank" rel="noopener">FiniteDifferences.jl</a> as a new primitive operation.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c">## FiniteDifferences</span>
+</span></span><span class="line"><span class="cl"><span class="k">struct</span> <span class="kt">FDMBackend</span><span class="p">{</span><span class="kt">A</span><span class="p">}</span> <span class="o">&lt;:</span> <span class="kt">AD</span><span class="o">.</span><span class="n">AbstractFiniteDifference</span>
+</span></span><span class="line"><span class="cl">    <span class="n">alg</span><span class="o">::</span><span class="kt">A</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="n">FDMBackend</span><span class="p">()</span> <span class="o">=</span> <span class="n">FDMBackend</span><span class="p">(</span><span class="n">central_fdm</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="k">const</span> <span class="n">fdm_backend</span> <span class="o">=</span> <span class="n">FDMBackend</span><span class="p">()</span>
+</span></span><span class="line"><span class="cl"><span class="c"># Minimal interface</span>
+</span></span><span class="line"><span class="cl"><span class="n">AD</span><span class="o">.</span><span class="nd">@primitive</span> <span class="k">function</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">ab</span><span class="o">::</span><span class="kt">FDMBackend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">xs</span><span class="o">...</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="n">FiniteDifferences</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">ab</span><span class="o">.</span><span class="n">alg</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">xs</span><span class="o">...</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># AD Jacobian returns tuple</span>
+</span></span><span class="line"><span class="cl"><span class="c"># df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]</span>
+</span></span><span class="line"><span class="cl"><span class="c"># df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">fdm_backend</span><span class="p">,</span> <span class="n">x</span><span class="o">-&gt;</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">),</span> <span class="n">x₀</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span> <span class="o">≈</span> <span class="n">Jacobian</span><span class="p">(</span><span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">fdm_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span> <span class="o">≈</span> <span class="n">Jacobian</span><span class="p">(</span><span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span></code></pre></div><p>After overloading the <code>step</code> function, we can run the Gauss-Newton algorithm as follows:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">step</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">backend</span><span class="p">,</span> <span class="n">α</span><span class="o">=</span><span class="mi">1</span><span class="o">//</span><span class="mi">1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">x2</span> <span class="o">=</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="k">while</span> <span class="o">!</span><span class="n">done</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">x2</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">J</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">    <span class="n">d</span> <span class="o">=</span> <span class="o">-</span><span class="n">inv</span><span class="p">(</span><span class="n">J</span><span class="o">&#39;*</span><span class="n">J</span><span class="p">)</span><span class="o">*</span><span class="n">J</span><span class="o">&#39;*</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">copyto!</span><span class="p">(</span><span class="n">x2</span><span class="p">,</span><span class="n">x</span> <span class="o">+</span> <span class="n">α</span><span class="o">*</span><span class="n">d</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">α</span> <span class="o">=</span> <span class="n">α</span><span class="o">//</span><span class="mi">2</span>
+</span></span><span class="line"><span class="cl">  <span class="k">end</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="n">x2</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">xs</span> <span class="o">=</span> <span class="p">[</span><span class="n">x₀</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">GaussNewton!</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="n">fdm_backend</span><span class="p">)</span>
+</span></span></code></pre></div><p>If we want to use reverse-mode AD instead, for example via <a href="https://github.com/FluxML/Zygote.jl" target="_blank" rel="noopener">Zygote.jl</a>, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c">## Zygote</span>
+</span></span><span class="line"><span class="cl"><span class="k">struct</span> <span class="kt">ZygoteBackend</span> <span class="o">&lt;:</span> <span class="kt">AD</span><span class="o">.</span><span class="n">AbstractReverseMode</span> <span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="k">const</span> <span class="n">zygote_backend</span> <span class="o">=</span> <span class="n">ZygoteBackend</span><span class="p">()</span>
+</span></span><span class="line"><span class="cl"><span class="n">AD</span><span class="o">.</span><span class="nd">@primitive</span> <span class="k">function</span> <span class="n">pullback_function</span><span class="p">(</span><span class="n">ab</span><span class="o">::</span><span class="kt">ZygoteBackend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">xs</span><span class="o">...</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="k">function</span> <span class="p">(</span><span class="n">vs</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="c"># Supports only single output</span>
+</span></span><span class="line"><span class="cl">        <span class="n">_</span><span class="p">,</span> <span class="n">back</span> <span class="o">=</span> <span class="n">Zygote</span><span class="o">.</span><span class="n">pullback</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">xs</span><span class="o">...</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="k">if</span> <span class="n">vs</span> <span class="k">isa</span> <span class="kt">AbstractVector</span>
+</span></span><span class="line"><span class="cl">            <span class="n">back</span><span class="p">(</span><span class="n">vs</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="k">else</span>
+</span></span><span class="line"><span class="cl">            <span class="nd">@assert</span> <span class="n">length</span><span class="p">(</span><span class="n">vs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
+</span></span><span class="line"><span class="cl">            <span class="n">back</span><span class="p">(</span><span class="n">vs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">        <span class="k">end</span>
+</span></span><span class="line"><span class="cl">    <span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="c">##</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">minimum</span><span class="p">(</span><span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">fdm_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">.≈</span> <span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">zygote_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">xs</span> <span class="o">=</span> <span class="p">[</span><span class="n">x₀</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">GaussNewton!</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="n">zygote_backend</span><span class="p">)</span>
+</span></span></code></pre></div><p>Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (<a href="https://github.com/JuliaDiff/ForwardDiff.jl" target="_blank" rel="noopener">ForwardDiff.jl</a>)</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c">## ForwardDiff</span>
+</span></span><span class="line"><span class="cl"><span class="k">struct</span> <span class="kt">ForwardDiffBackend</span> <span class="o">&lt;:</span> <span class="kt">AD</span><span class="o">.</span><span class="n">AbstractForwardMode</span> <span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="k">const</span> <span class="n">forwarddiff_backend</span> <span class="o">=</span> <span class="n">ForwardDiffBackend</span><span class="p">()</span>
+</span></span><span class="line"><span class="cl"><span class="n">AD</span><span class="o">.</span><span class="nd">@primitive</span> <span class="k">function</span> <span class="n">pushforward_function</span><span class="p">(</span><span class="n">ab</span><span class="o">::</span><span class="kt">ForwardDiffBackend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">xs</span><span class="o">...</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="c"># jvp = f&#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="k">function</span> <span class="p">(</span><span class="n">vs</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="k">if</span> <span class="n">xs</span> <span class="k">isa</span> <span class="kt">Tuple</span>
+</span></span><span class="line"><span class="cl">            <span class="nd">@assert</span> <span class="n">length</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">2</span>
+</span></span><span class="line"><span class="cl">            <span class="k">if</span> <span class="n">length</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span>
+</span></span><span class="line"><span class="cl">                <span class="p">(</span><span class="n">ForwardDiff</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">h</span><span class="o">-&gt;</span><span class="n">f</span><span class="p">(</span><span class="n">xs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">h</span><span class="o">*</span><span class="n">vs</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span><span class="mi">0</span><span class="p">),)</span>
+</span></span><span class="line"><span class="cl">            <span class="k">else</span>
+</span></span><span class="line"><span class="cl">                <span class="n">ForwardDiff</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">h</span><span class="o">-&gt;</span><span class="n">f</span><span class="p">(</span><span class="n">xs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">h</span><span class="o">*</span><span class="n">vs</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">xs</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">+</span><span class="n">h</span><span class="o">*</span><span class="n">vs</span><span class="p">[</span><span class="mi">2</span><span class="p">]),</span><span class="mi">0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">            <span class="k">end</span>
+</span></span><span class="line"><span class="cl">        <span class="k">else</span>
+</span></span><span class="line"><span class="cl">            <span class="n">ForwardDiff</span><span class="o">.</span><span class="n">derivative</span><span class="p">(</span><span class="n">h</span><span class="o">-&gt;</span><span class="n">f</span><span class="p">(</span><span class="n">xs</span><span class="o">+</span><span class="n">h</span><span class="o">*</span><span class="n">vs</span><span class="p">),</span><span class="mi">0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="k">end</span>
+</span></span><span class="line"><span class="cl">    <span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="c">##</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">minimum</span><span class="p">(</span><span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">fdm_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">.≈</span> <span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">forwarddiff_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">xs</span> <span class="o">=</span> <span class="p">[</span><span class="n">x₀</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">GaussNewton!</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="n">forwarddiff_backend</span><span class="p">)</span>
+</span></span></code></pre></div><p>where we have used that the Jacobian-vector product $f&rsquo;(x)v$, i.e., the primitives of forward-mode AD, can be computed by <a href="https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12" target="_blank" rel="noopener">differentiating $f(x + hv)$ with respect to $h$ at 0</a>.</p>
+<p>Many AD packages, such as Zygote, have troubles with mutating functions. <a href="https://github.com/wsmoses/Enzyme.jl" target="_blank" rel="noopener">Enzyme.jl</a> is one of the exceptions. Additionally, it is very fast and has further improved the performance of the <a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944" target="_blank" rel="noopener">adjoints implemented within the DiffEqSensitivity package</a>.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c">## Enzyme</span>
+</span></span><span class="line"><span class="cl"><span class="k">struct</span> <span class="kt">EnzymeBackend</span><span class="p">{</span><span class="kt">T1</span><span class="p">,</span><span class="kt">T2</span><span class="p">,</span><span class="kt">T3</span><span class="p">,</span><span class="kt">T4</span><span class="p">}</span> <span class="o">&lt;:</span> <span class="kt">AD</span><span class="o">.</span><span class="n">AbstractReverseMode</span>
+</span></span><span class="line"><span class="cl">    <span class="n">out</span><span class="o">::</span><span class="kt">T1</span>
+</span></span><span class="line"><span class="cl">    <span class="n">λ</span><span class="o">::</span><span class="kt">T2</span>
+</span></span><span class="line"><span class="cl">    <span class="n">∂f_∂x</span><span class="o">::</span><span class="kt">T3</span>
+</span></span><span class="line"><span class="cl">    <span class="n">∂f_∂p</span><span class="o">::</span><span class="kt">T4</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">out</span> <span class="o">=</span> <span class="n">zero</span><span class="p">(</span><span class="n">x₀</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">λ</span> <span class="o">=</span> <span class="n">zero</span><span class="p">(</span><span class="n">x₀</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">∂f_∂x</span> <span class="o">=</span> <span class="n">zero</span><span class="p">(</span><span class="n">x₀</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">∂f_∂p</span> <span class="o">=</span> <span class="n">zero</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">const</span> <span class="n">enzyme_backend</span> <span class="o">=</span> <span class="n">EnzymeBackend</span><span class="p">(</span><span class="n">out</span><span class="p">,</span><span class="n">λ</span><span class="p">,</span><span class="n">∂f_∂x</span><span class="p">,</span><span class="n">∂f_∂p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">AD</span><span class="o">.</span><span class="nd">@primitive</span> <span class="k">function</span> <span class="n">pullback_function</span><span class="p">(</span><span class="n">ab</span><span class="o">::</span><span class="kt">EnzymeBackend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">xs</span><span class="o">...</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="k">function</span> <span class="p">(</span><span class="n">vs</span><span class="p">)</span>  
+</span></span><span class="line"><span class="cl">        <span class="c"># enzyme works only with inplace functions</span>
+</span></span><span class="line"><span class="cl">        <span class="k">if</span> <span class="o">!</span><span class="p">(</span><span class="n">vs</span> <span class="k">isa</span> <span class="kt">AbstractVector</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">            <span class="nd">@assert</span> <span class="n">length</span><span class="p">(</span><span class="n">vs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span> <span class="c"># Supports only single output</span>
+</span></span><span class="line"><span class="cl">            <span class="n">vs</span> <span class="o">=</span> <span class="n">vs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">        <span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">        <span class="k">if</span> <span class="n">xs</span> <span class="k">isa</span> <span class="kt">Tuple</span>
+</span></span><span class="line"><span class="cl">            <span class="nd">@assert</span> <span class="n">length</span><span class="p">(</span><span class="n">xs</span><span class="p">)</span> <span class="o">==</span> <span class="mi">2</span>  <span class="c"># hard-coded for use case with two inputs</span>
+</span></span><span class="line"><span class="cl">            <span class="n">x₀</span> <span class="o">=</span> <span class="n">xs</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">            <span class="n">p</span> <span class="o">=</span> <span class="n">xs</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">        <span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">        <span class="nd">@unpack</span> <span class="n">out</span><span class="p">,</span> <span class="n">λ</span><span class="p">,</span> <span class="n">∂f_∂x</span><span class="p">,</span> <span class="n">∂f_∂p</span> <span class="o">=</span> <span class="n">ab</span> <span class="c"># cached in the struct, could also be created in here</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">        <span class="n">∂f_∂x</span> <span class="o">.*=</span> <span class="nb">false</span>
+</span></span><span class="line"><span class="cl">        <span class="n">∂f_∂p</span> <span class="o">.*=</span> <span class="nb">false</span>
+</span></span><span class="line"><span class="cl">        <span class="n">out</span> <span class="o">.*=</span> <span class="nb">false</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">        <span class="n">copyto!</span><span class="p">(</span><span class="n">λ</span><span class="p">,</span> <span class="n">vs</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">        <span class="n">autodiff</span><span class="p">(</span><span class="n">Duplicated</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">λ</span><span class="p">),</span> <span class="n">Duplicated</span><span class="p">(</span><span class="n">x₀</span><span class="p">,</span> <span class="n">∂f_∂x</span><span class="p">),</span> <span class="n">Duplicated</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">∂f_∂p</span><span class="p">))</span> <span class="k">do</span> <span class="n">_out</span><span class="p">,</span><span class="n">_x</span><span class="p">,</span> <span class="n">_p</span>
+</span></span><span class="line"><span class="cl">            <span class="n">f</span><span class="p">(</span><span class="n">_out</span><span class="p">,</span><span class="n">_x</span><span class="p">,</span><span class="n">_p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="k">end</span>
+</span></span><span class="line"><span class="cl">        <span class="k">return</span> <span class="p">(</span><span class="n">∂f_∂x</span><span class="p">,</span><span class="n">∂f_∂p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="n">AD</span><span class="o">.</span><span class="n">isinplace</span><span class="p">(</span><span class="n">ab</span><span class="o">::</span><span class="kt">EnzymeBackend</span><span class="p">)</span> <span class="o">=</span> <span class="nb">true</span>
+</span></span><span class="line"><span class="cl"><span class="n">AD</span><span class="o">.</span><span class="n">primalvalue</span><span class="p">(</span><span class="n">ab</span><span class="o">::</span><span class="kt">EnzymeBackend</span><span class="p">,</span> <span class="nb">nothing</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">xs</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">ab</span><span class="o">.</span><span class="n">out</span><span class="p">,</span><span class="n">xs</span><span class="o">...</span><span class="p">);</span><span class="k">return</span> <span class="n">ab</span><span class="o">.</span><span class="n">out</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="c">##</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">minimum</span><span class="p">(</span><span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">fdm_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="o">.≈</span> <span class="n">AD</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">enzyme_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">xs</span> <span class="o">=</span> <span class="p">[</span><span class="n">x₀</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">GaussNewton!</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="n">enzyme_backend</span><span class="p">)</span>
+</span></span></code></pre></div><p>Note that we have declared the Enzyme backend as <code>inplace</code> (which is important for internal control flow) and specified a <code>primalvalue</code> function returning the primal value of the forward pass.</p>
 <h2 id="some-current-glitches">Some current glitches</h2>
 <p>First, the push forward of a tuple of vectors, e.g., $(v_1, v_2)$, for a function with several input arguments is currently ambiguous. While <code>AD.jacobian</code> primitives and <code>AD.pullback_function</code> primitives interpret the push forward of our $f$ function as</p>
 <p>$$
@@ -987,86 +1147,77 @@ <h2 id="some-current-glitches">Some current glitches</h2>
 <p>$$
 \frac{\partial f(x_0,p)}{\partial x} v_1 + \frac{\partial f(x_0,p)}{\partial p} v_2.
 $$</p>
-<pre><code class="language-julia"># pushforward_function wrt to multiple vectors is currently ambiguous
-vs = (randn(2), randn(2))
-res1 = AD.pushforward_function(fdm_backend, f, x₀, p)(vs)
-res2 = AD.pushforward_function(forwarddiff_backend, f, x₀, p)(vs)
-
-@test res2 ≈ res1[1] + res1[2]
-</code></pre>
-<p>Thus, we currently solve this issue by augmenting the input in the case of <code>AD.pushforward_function</code> primitives.</p>
-<pre><code class="language-julia">res2a = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((vs[1], zero(vs[2])))
-res2b = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((zero(vs[1]), vs[2]))
-
-@test res2a ≈ res1[1]
-@test res2b ≈ res1[2]
-</code></pre>
-<p>The plural &ldquo;primitives&rdquo; is used here because we may have different <code>pushforward_function</code> primitives for different backends. For instance, we can define an additional <code>pushforward_function</code> primitive for FiniteDifferences by:</p>
-<pre><code class="language-julia">struct FDMBackend2{A} &lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend2() = FDMBackend2(central_fdm(5, 1))
-const fdm_backend2 = FDMBackend2()
-AD.@primitive function pushforward_function(ab::FDMBackend2, f, xs...)
-    return function (vs)
-        FDM.jvp(ab.alg, f, tuple.(xs, vs)...)
-    end
-end
-</code></pre>
-<p>Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the <code>Hessian</code> function.</p>
-<pre><code class="language-julia"># Hessian only defined with respect to single input variable
-@test_throws AssertionError H1 = AD.hessian(forwarddiff_backend, g, x₀, p)
-H1 = AD.hessian(forwarddiff_backend, x-&gt;g(x,p), x₀)
-H2 = AD.hessian(forwarddiff_backend, p-&gt;g(x₀,p), p)
-</code></pre>
-<p>Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a <code>HigherOrderBackend</code> that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).</p>
-<pre><code class="language-julia"># Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)
-backends = AD.HigherOrderBackend((forwarddiff_backend,zygote_backend))
-H3 = AD.hessian(backends, x-&gt;g(x,p), x₀)
-</code></pre>
-<h2 id="outlook">Outlook</h2>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># pushforward_function wrt to multiple vectors is currently ambiguous</span>
+</span></span><span class="line"><span class="cl"><span class="n">vs</span> <span class="o">=</span> <span class="p">(</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">res1</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">pushforward_function</span><span class="p">(</span><span class="n">fdm_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)(</span><span class="n">vs</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">res2</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">pushforward_function</span><span class="p">(</span><span class="n">forwarddiff_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)(</span><span class="n">vs</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">res2</span> <span class="o">≈</span> <span class="n">res1</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">res1</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span></code></pre></div><p>Thus, we currently solve this issue by augmenting the input in the case of <code>AD.pushforward_function</code> primitives.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="n">res2a</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">pushforward_function</span><span class="p">(</span><span class="n">forwarddiff_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)((</span><span class="n">vs</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">zero</span><span class="p">(</span><span class="n">vs</span><span class="p">[</span><span class="mi">2</span><span class="p">])))</span>
+</span></span><span class="line"><span class="cl"><span class="n">res2b</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">pushforward_function</span><span class="p">(</span><span class="n">forwarddiff_backend</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)((</span><span class="n">zero</span><span class="p">(</span><span class="n">vs</span><span class="p">[</span><span class="mi">1</span><span class="p">]),</span> <span class="n">vs</span><span class="p">[</span><span class="mi">2</span><span class="p">]))</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">res2a</span> <span class="o">≈</span> <span class="n">res1</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">res2b</span> <span class="o">≈</span> <span class="n">res1</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span></code></pre></div><p>The plural &ldquo;primitives&rdquo; is used here because we may have different <code>pushforward_function</code> primitives for different backends. For instance, we can define an additional <code>pushforward_function</code> primitive for FiniteDifferences by:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">struct</span> <span class="kt">FDMBackend2</span><span class="p">{</span><span class="kt">A</span><span class="p">}</span> <span class="o">&lt;:</span> <span class="kt">AD</span><span class="o">.</span><span class="n">AbstractFiniteDifference</span>
+</span></span><span class="line"><span class="cl">    <span class="n">alg</span><span class="o">::</span><span class="kt">A</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="n">FDMBackend2</span><span class="p">()</span> <span class="o">=</span> <span class="n">FDMBackend2</span><span class="p">(</span><span class="n">central_fdm</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="k">const</span> <span class="n">fdm_backend2</span> <span class="o">=</span> <span class="n">FDMBackend2</span><span class="p">()</span>
+</span></span><span class="line"><span class="cl"><span class="n">AD</span><span class="o">.</span><span class="nd">@primitive</span> <span class="k">function</span> <span class="n">pushforward_function</span><span class="p">(</span><span class="n">ab</span><span class="o">::</span><span class="kt">FDMBackend2</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">xs</span><span class="o">...</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="k">function</span> <span class="p">(</span><span class="n">vs</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="n">FDM</span><span class="o">.</span><span class="n">jvp</span><span class="p">(</span><span class="n">ab</span><span class="o">.</span><span class="n">alg</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">tuple</span><span class="o">.</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">vs</span><span class="p">)</span><span class="o">...</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><p>Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the <code>Hessian</code> function.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># Hessian only defined with respect to single input variable</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@test_throws</span> <span class="kt">AssertionError</span> <span class="n">H1</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">hessian</span><span class="p">(</span><span class="n">forwarddiff_backend</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">x₀</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">H1</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">hessian</span><span class="p">(</span><span class="n">forwarddiff_backend</span><span class="p">,</span> <span class="n">x</span><span class="o">-&gt;</span><span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">),</span> <span class="n">x₀</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">H2</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">hessian</span><span class="p">(</span><span class="n">forwarddiff_backend</span><span class="p">,</span> <span class="n">p</span><span class="o">-&gt;</span><span class="n">g</span><span class="p">(</span><span class="n">x₀</span><span class="p">,</span><span class="n">p</span><span class="p">),</span> <span class="n">p</span><span class="p">)</span>
+</span></span></code></pre></div><p>Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a <code>HigherOrderBackend</code> that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)</span>
+</span></span><span class="line"><span class="cl"><span class="n">backends</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">HigherOrderBackend</span><span class="p">((</span><span class="n">forwarddiff_backend</span><span class="p">,</span><span class="n">zygote_backend</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">H3</span> <span class="o">=</span> <span class="n">AD</span><span class="o">.</span><span class="n">hessian</span><span class="p">(</span><span class="n">backends</span><span class="p">,</span> <span class="n">x</span><span class="o">-&gt;</span><span class="n">g</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">p</span><span class="p">),</span> <span class="n">x₀</span><span class="p">)</span>
+</span></span></code></pre></div><h2 id="outlook">Outlook</h2>
 <p>There are many other use cases, e.g.,</p>
 <ul>
-<li>
-<a href="https://diffeq.sciml.ai/stable/analysis/sensitivity/" target="_blank" rel="noopener">Sensitivity analysis of differential equations</a> requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.</li>
-<li>The 
-<a href="https://en.wikipedia.org/wiki/Newton%27s_method" target="_blank" rel="noopener">Newton–Raphson method</a> for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.</li>
-<li>The 
-<a href="https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization" target="_blank" rel="noopener">Newton method</a> in optimization requires the computation of the Hessian.</li>
+<li><a href="https://diffeq.sciml.ai/stable/analysis/sensitivity/" target="_blank" rel="noopener">Sensitivity analysis of differential equations</a> requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.</li>
+<li>The <a href="https://en.wikipedia.org/wiki/Newton%27s_method" target="_blank" rel="noopener">Newton–Raphson method</a> for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.</li>
+<li>The <a href="https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization" target="_blank" rel="noopener">Newton method</a> in optimization requires the computation of the Hessian.</li>
 </ul>
 <p>AbstractDifferentiation.jl is by no means complete yet. We are still in the very early stages, but we hope to make significant progress in the coming weeks. Some of the next steps are:</p>
 <ul>
 <li>fixing remaining bugs, e.g., with respect to the computation of the Hessian and</li>
-<li>adding AD/Finite Differentiation packages such as 
-<a href="https://github.com/JuliaDiff/Diffractor.jl" target="_blank" rel="noopener">Diffractor</a>.</li>
+<li>adding AD/Finite Differentiation packages such as <a href="https://github.com/JuliaDiff/Diffractor.jl" target="_blank" rel="noopener">Diffractor</a>.</li>
 </ul>
 <p>If you have any questions or comments, please don’t hesitate to contact me!</p>
-<section class="footnotes" role="doc-endnotes">
+<div class="footnotes" role="doc-endnotes">
 <hr>
 <ol>
-<li id="fn:1" role="doc-endnote">
-<p>Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019). <a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:1">
+<p>Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019).&#160;<a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:2" role="doc-endnote">
-<p>Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012). <a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:2">
+<p>Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012).&#160;<a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:3" role="doc-endnote">
-<p>Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019). <a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:3">
+<p>Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019).&#160;<a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:4" role="doc-endnote">
-<p>Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). <a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:4">
+<p>Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&#160;<a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:5" role="doc-endnote">
-<p>Raj Dandekar, Chris Rackauckas, et al., Patterns <strong>1</strong>, 100145 (2020). <a href="#fnref:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:5">
+<p>Raj Dandekar, Chris Rackauckas, et al., Patterns <strong>1</strong>, 100145 (2020).&#160;<a href="#fnref:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:6" role="doc-endnote">
-<p>Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. <strong>1</strong>, 035009 (2020). <a href="#fnref:6" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:6">
+<p>Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. <strong>1</strong>, 035009 (2020).&#160;<a href="#fnref:6" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:7" role="doc-endnote">
-<p>Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. <strong>2</strong>, 035004 (2021). <a href="#fnref:7" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:7">
+<p>Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. <strong>2</strong>, 035004 (2021).&#160;<a href="#fnref:7" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
 </ol>
-</section>
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-  
-  <span>Frank Schäfer and Moritz Schauer</span>
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+      Frank Schäfer and Moritz Schauer</span>
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       <p>In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup>. We highlight differences between explicit<sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> and implicit events<sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup> <sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>. As a paradigmatic example, we consider a bouncing ball described by the ODE</p>
 <p>$$
 \begin{aligned}
-\text{d}z(t) &amp;= v(t) \text{d}t, \\<br>
+\text{d}z(t) &amp;= v(t) \text{d}t, \\
 \text{d}v(t) &amp;= -\mathrm g\thinspace  \text{d}t
 \end{aligned}<br>
 $$</p>
 <p>with initial condition</p>
 <p>$$
 \begin{aligned}
-z(t=0) &amp;= z_0 = 5, \\<br>
+z(t=0) &amp;= z_0 = 5, \\
 v(t=0) &amp;= v_0 = -0.1.
 \end{aligned}<br>
 $$</p>
 <p>The initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.</p>
-<p>Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (
-<a href="https://en.wikipedia.org/wiki/Coefficient_of_restitution" target="_blank" rel="noopener">coefficient of restitution</a>) that accounts for a imperfect elastic bounce on the ground.</p>
+<p>Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (<a href="https://en.wikipedia.org/wiki/Coefficient_of_restitution" target="_blank" rel="noopener">coefficient of restitution</a>) that accounts for a imperfect elastic bounce on the ground.</p>
 <p>When ignoring the bounces, we can straightforwardly integrate the ODE analytically</p>
 <p>$$
 \begin{aligned}
-z(t) &amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\<br>
+z(t) &amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\
 v(t) &amp;= v_0 - \mathrm g\thinspace  t
 \end{aligned}<br>
 $$</p>
-<p>or numerically using the OrdinaryDiffEq package from the 
-<a href="https://sciml.ai/" target="_blank" rel="noopener">SciML</a> ecosystem.</p>
-<pre><code class="language-julia">### simulate forward
+<p>or numerically using the OrdinaryDiffEq package from the <a href="https://sciml.ai/" target="_blank" rel="noopener">SciML</a> ecosystem.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c">### simulate forward</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">ForwardDiff</span><span class="p">,</span> <span class="n">Zygote</span><span class="p">,</span> <span class="n">OrdinaryDiffEq</span><span class="p">,</span> <span class="n">DiffEqSensitivity</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Plots</span><span class="p">,</span> <span class="n">LaTeXStrings</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># dynamics</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># parameters and solve</span>
+</span></span><span class="line"><span class="cl"><span class="n">z0</span> <span class="o">=</span> <span class="mf">5.0</span>
+</span></span><span class="line"><span class="cl"><span class="n">v0</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.1</span>
+</span></span><span class="line"><span class="cl"><span class="n">t0</span> <span class="o">=</span> <span class="mf">0.0</span>
+</span></span><span class="line"><span class="cl"><span class="n">tend</span> <span class="o">=</span> <span class="mf">1.9</span>
+</span></span><span class="line"><span class="cl"><span class="n">g</span> <span class="o">=</span> <span class="mi">10</span>
+</span></span><span class="line"><span class="cl"><span class="n">γ</span> <span class="o">=</span> <span class="mf">0.8</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">u0</span> <span class="o">=</span> <span class="p">[</span><span class="n">z0</span><span class="p">,</span><span class="n">v0</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">tspan</span> <span class="o">=</span> <span class="p">(</span><span class="n">t0</span><span class="p">,</span><span class="n">tend</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="p">,</span> <span class="n">γ</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">u0</span><span class="p">,</span><span class="n">tspan</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># plot forward trajectory</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">saveat</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#34;z(t)&#34;</span> <span class="s">&#34;v(t)&#34;</span><span class="p">],</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;t&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="ss">:bottomleft</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">hline!</span><span class="p">(</span><span class="n">pl</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#34;black&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">savefig</span><span class="p">(</span><span class="n">pl</span><span class="p">,</span><span class="s">&#34;BB_forward_no_bounce.png&#34;</span><span class="p">)</span>
+</span></span></code></pre></div><p>
+
+
+
 
-using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity
-using Plots, LaTeXStrings
 
-# dynamics
-function f(du,u,p,t)
-  du[1] = u[2]
-  du[2] = -p[1]
-end
 
-# parameters and solve
-z0 = 5.0
-v0 = -0.1
-t0 = 0.0
-tend = 1.9
-g = 10
-γ = 0.8
 
-u0 = [z0,v0]
-tspan = (t0,tend)
-p = [g, γ]
-prob = ODEProblem(f,u0,tspan,p)
 
-# plot forward trajectory
-sol = solve(prob,Tsit5(),saveat=0.1)
-pl = plot(sol, label = [&quot;z(t)&quot; &quot;v(t)&quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &quot;t&quot;, legend=:bottomleft)
-hline!(pl, [0.0], label=false, color=&quot;black&quot;)
-savefig(pl,&quot;BB_forward_no_bounce.png&quot;)
-</code></pre>
-<p><img src="https://i.imgur.com/fKTiDEe.png" alt=""></p>
 
 
 
 
 
-  
 
 
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="https://i.imgur.com/fKTiDEe.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
+</p>
 
 
 
@@ -738,25 +946,26 @@ <h1>Sensitivity Analysis of Hybrid Differential Equations</h1>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/BB_forward_no_bounce.png" >
 
 
-  <img src="/img/BB_forward_no_bounce.png" alt=""  >
-</a>
 
 
 
-</figure>
+
+
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="BB_forward_no_bounce.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>Of course, this way the ball continues to fall through the barrier at $z=0$.</p>
 <h2 id="forward-simulation-with-events">Forward simulation with events</h2>
 <p>At time $\tau$ around $\tau \approx 1$, the ball hits the ground $z(\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\tau$ with new initial conditions</p>
 <p>$$
 \begin{aligned}<br>
-z({\tau}) &amp;=  z(\tau-) ,\\<br>
+z({\tau}) &amp;=  z(\tau-) ,\\
 v({\tau})&amp;= -\gamma v(\tau-) ,
 \end{aligned}<br>
 $$</p>
@@ -773,15 +982,15 @@ <h3 id="explicit-events">Explicit events</h3>
 <p>The full explicit trajectory $z_{\rm exp}(t) = z(t)$ is determined by</p>
 <p>$$
 z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp; \forall t &lt; \tau, \\<br>
--0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\<br>
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp; \forall t &lt; \tau, \\
+-0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\
 \quad  - (-0.99005 + t) (-0.99005 \mathrm g + v_0)\gamma  ,&amp; \forall t \ge \tau,
 \end{cases}
 $$</p>
 <p>where we used</p>
 <p>$$
 \begin{aligned}<br>
-z({\tau})&amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\<br>
+z({\tau})&amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\
 v({\tau})&amp;= -\gamma v({\tau-}) = -\gamma(v_0 - 0.99005 \mathrm g)  .
 \end{aligned}<br>
 $$</p>
@@ -790,15 +999,14 @@ <h3 id="explicit-events">Explicit events</h3>
 a(z,v) = (z, -\gamma v).
 $$</p>
 <p>Numerically, we use a <code>DiscreteCallback</code> in this case to simulate the system.</p>
-<pre><code class="language-julia"># solve with DiscreteCallback (explicit event)
-tstar = (v0 + sqrt(v0^2+2*z0*g))/g
-condition1(u,t,integrator) = (t == tstar)
-affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2]
-cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true))
-
-sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar])
-</code></pre>
-<p>Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># solve with DiscreteCallback (explicit event)</span>
+</span></span><span class="line"><span class="cl"><span class="n">tstar</span> <span class="o">=</span> <span class="p">(</span><span class="n">v0</span> <span class="o">+</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">v0</span><span class="o">^</span><span class="mi">2</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">z0</span><span class="o">*</span><span class="n">g</span><span class="p">))</span><span class="o">/</span><span class="n">g</span>
+</span></span><span class="line"><span class="cl"><span class="n">condition1</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">integrator</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">t</span> <span class="o">==</span> <span class="n">tstar</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">affect!</span><span class="p">(</span><span class="n">integrator</span><span class="p">)</span> <span class="o">=</span> <span class="n">integrator</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">integrator</span><span class="o">.</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">integrator</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">cb1</span> <span class="o">=</span> <span class="n">DiscreteCallback</span><span class="p">(</span><span class="n">condition1</span><span class="p">,</span><span class="n">affect!</span><span class="p">,</span><span class="n">save_positions</span><span class="o">=</span><span class="p">(</span><span class="nb">true</span><span class="p">,</span><span class="nb">true</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">sol1</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">callback</span><span class="o">=</span><span class="n">cb1</span><span class="p">,</span> <span class="n">saveat</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">tstops</span><span class="o">=</span><span class="p">[</span><span class="n">tstar</span><span class="p">])</span>
+</span></span></code></pre></div><p>Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.</p>
 <h3 id="implicit-events">Implicit events</h3>
 <p>The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)</p>
 <p>$$
@@ -807,7 +1015,7 @@ <h3 id="implicit-events">Implicit events</h3>
 <p>where an event occurs at time $\tau$ if $g(z(\tau),v(\tau),p,\tau) = 0$. We have already used this condition to define our impact time $\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time  $g(z,v,p,t) = 0$.</p>
 <p>As in the previous case, we can analytically compute the full trajectory of the ball. By substituting the formula for $\tau$ we have at the event time</p>
 <p>\begin{aligned}
-z({\tau})&amp;= 0, \\<br>
+z({\tau})&amp;= 0, \\
 v({\tau}-)&amp;= - \sqrt{v_0^2 + 2 \mathrm g z_0}
 \end{aligned}</p>
 <p>for the left limit and</p>
@@ -817,34 +1025,26 @@ <h3 id="implicit-events">Implicit events</h3>
 <p>right after the bounce. Thus, the full trajectory $z_{\rm imp}(t) = z(t)$ is given by</p>
 <p>$$
 (\star) \quad z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp; \forall t &lt; \tau ,\\<br>
--\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\<br>
-\quad\cdot \space (-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0} (1 + 2 \gamma)), &amp; \forall t \ge \tau.
-\end{cases}
-$$</p>
-<p>This is correct even if one substitutes, e.g., a value with higher precision $\mathrm g = 9.81$ for the gravitation constant.</p>
-<p>Numerically, we use a <code>ContinuousCallback</code> in this case.</p>
-<pre><code class="language-julia"># solve with ContinuousCallback (implicit event)
-condition2(u,t,integrator) = u[1] # Event happens when condition2(u,t,integrator) == 0
-cb2 = ContinuousCallback(condition2,affect!,save_positions=(true,true))
-sol2 = solve(prob,Tsit5(),callback=cb2,saveat=0.1)
-</code></pre>
-<p>We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).</p>
-<pre><code class="language-julia"># plot forward trajectory
-pl1 = plot(sol1, label = [&quot;z(t)&quot; &quot;v(t)&quot;], title=&quot;explicit event&quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &quot;t&quot;, legend=:bottomright)
-pl2 = plot(sol2, label = [&quot;z(t)&quot; &quot;v(t)&quot;], title=&quot;implicit event&quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &quot;t&quot;, legend=:bottomright)
-hline!(pl1, [0.0], label=false, color=&quot;black&quot;)
-hline!(pl2, [0.0], label=false, color=&quot;black&quot;)
-pl = plot(pl1,pl2)
-savefig(pl,&quot;BB_forward.png&quot;)
-</code></pre>
-
-
-
-
-
-  
-
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp; \forall t &lt; \tau ,\\
+-\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\
+\quad\cdot \space (-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0} (1 + 2 \gamma)), &amp; \forall t \ge \tau.
+\end{cases}
+$$</p>
+<p>This is correct even if one substitutes, e.g., a value with higher precision $\mathrm g = 9.81$ for the gravitation constant.</p>
+<p>Numerically, we use a <code>ContinuousCallback</code> in this case.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># solve with ContinuousCallback (implicit event)</span>
+</span></span><span class="line"><span class="cl"><span class="n">condition2</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">integrator</span><span class="p">)</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="c"># Event happens when condition2(u,t,integrator) == 0</span>
+</span></span><span class="line"><span class="cl"><span class="n">cb2</span> <span class="o">=</span> <span class="n">ContinuousCallback</span><span class="p">(</span><span class="n">condition2</span><span class="p">,</span><span class="n">affect!</span><span class="p">,</span><span class="n">save_positions</span><span class="o">=</span><span class="p">(</span><span class="nb">true</span><span class="p">,</span><span class="nb">true</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol2</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">callback</span><span class="o">=</span><span class="n">cb2</span><span class="p">,</span><span class="n">saveat</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
+</span></span></code></pre></div><p>We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># plot forward trajectory</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol1</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#34;z(t)&#34;</span> <span class="s">&#34;v(t)&#34;</span><span class="p">],</span> <span class="n">title</span><span class="o">=</span><span class="s">&#34;explicit event&#34;</span><span class="p">,</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;t&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="ss">:bottomright</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl2</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol2</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#34;z(t)&#34;</span> <span class="s">&#34;v(t)&#34;</span><span class="p">],</span> <span class="n">title</span><span class="o">=</span><span class="s">&#34;implicit event&#34;</span><span class="p">,</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;t&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="ss">:bottomright</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">hline!</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#34;black&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">hline!</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#34;black&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span><span class="n">pl2</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">savefig</span><span class="p">(</span><span class="n">pl</span><span class="p">,</span><span class="s">&#34;BB_forward.png&#34;</span><span class="p">)</span>
+</span></span></code></pre></div>
 
 
 
@@ -855,18 +1055,17 @@ <h3 id="implicit-events">Implicit events</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/BB_forward.png" >
 
 
-  <img src="/img/BB_forward.png" alt=""  >
-</a>
 
 
 
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="BB_forward.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>In addition, the implicitly defined impact time via the <code>ContinuousCallback</code> also changes appropriately when changing the initial conditions or the parameters, for example when using  $\mathrm g = 9.81$ for the gravitation constant. In other words, the event time $\tau=\tau(p,z_0,v_0,t_0)$ is a function of the parameters and initial conditions, and is implicitly defined by the event condition.</p>
 <p>Suppose we let the ball drop from a somewhat higher position now. Does an increase in height $z$ at $t=0$ give an increase or decrease in height at the end time $t_\text{end}=1.9$? This is something we can answer with sensitivity analysis. For example if we increase the height by (a fraction of) one unit then using $(\star)$</p>
@@ -875,36 +1074,30 @@ <h3 id="implicit-events">Implicit events</h3>
 $$</p>
 <p>meaning the height at $t_\text{end}$ is also by a corresponding fraction of 0.84 units higher.</p>
 <p>We can verify this visually:</p>
-<pre><code class="language-julia"># animate forward trajectory
-sol3 = solve(remake(prob,u0=[u0[1]+0.5,u0[2]]),Tsit5(),callback=cb2,saveat=0.01)
-
-plt2 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &quot;t&quot;, legend=:bottomright, color=&quot;black&quot;, xlims=(t0,tend))
-hline!(plt2, [0.0], label=false, color=&quot;black&quot;)
-plot!(plt2, sol3, tspan=(t0,tend), color=[1 2], label = [&quot;z(t)&quot; &quot;v(t)&quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &quot;t&quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)
-
-list_plots = []
-for t in sol3.t
-  tstart = 0.0
-
-  plt1 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &quot;t&quot;, legend=:bottomright, color=&quot;black&quot;)
-  hline!(plt1, [0.0], label=false, color=&quot;black&quot;)
-  plot!(plt1, sol3, tspan=(t0,t), color=[1 2], label = [&quot;z(t)&quot; &quot;v(t)&quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &quot;t&quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-  scatter!(plt1,[t,t], sol3(t), color=[1, 2], label=false)
-  plt = plot(plt1,plt2)
-  push!(list_plots, plt)
-end
-
-plot(list_plots[100])
-
-anim = animate(list_plots,every=1)
-</code></pre>
-
-
-
-
-
-  
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># animate forward trajectory</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol3</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">remake</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">u0</span><span class="o">=</span><span class="p">[</span><span class="n">u0</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="mf">0.5</span><span class="p">,</span><span class="n">u0</span><span class="p">[</span><span class="mi">2</span><span class="p">]]),</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">callback</span><span class="o">=</span><span class="n">cb2</span><span class="p">,</span><span class="n">saveat</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">plt2</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol2</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="nb">false</span><span class="p">,</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;t&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="ss">:bottomright</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#34;black&#34;</span><span class="p">,</span> <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span><span class="n">tend</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">hline!</span><span class="p">(</span><span class="n">plt2</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#34;black&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">plot!</span><span class="p">(</span><span class="n">plt2</span><span class="p">,</span> <span class="n">sol3</span><span class="p">,</span> <span class="n">tspan</span><span class="o">=</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span><span class="n">tend</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span> <span class="mi">2</span><span class="p">],</span> <span class="n">label</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#34;z(t)&#34;</span> <span class="s">&#34;v(t)&#34;</span><span class="p">],</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;t&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="ss">:bottomright</span><span class="p">,</span> <span class="n">denseplot</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span> <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span><span class="n">tend</span><span class="p">),</span> <span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">11</span><span class="p">,</span><span class="mi">9</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="c"># scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">list_plots</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl"><span class="k">for</span> <span class="n">t</span> <span class="k">in</span> <span class="n">sol3</span><span class="o">.</span><span class="n">t</span>
+</span></span><span class="line"><span class="cl">  <span class="n">tstart</span> <span class="o">=</span> <span class="mf">0.0</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">plt1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol2</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="nb">false</span><span class="p">,</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;t&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="ss">:bottomright</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#34;black&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">hline!</span><span class="p">(</span><span class="n">plt1</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#34;black&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">plot!</span><span class="p">(</span><span class="n">plt1</span><span class="p">,</span> <span class="n">sol3</span><span class="p">,</span> <span class="n">tspan</span><span class="o">=</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span><span class="n">t</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span> <span class="mi">2</span><span class="p">],</span> <span class="n">label</span> <span class="o">=</span> <span class="p">[</span><span class="s">&#34;z(t)&#34;</span> <span class="s">&#34;v(t)&#34;</span><span class="p">],</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">legendfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;t&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="ss">:bottomright</span><span class="p">,</span> <span class="n">denseplot</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span> <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span><span class="n">tend</span><span class="p">),</span> <span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">11</span><span class="p">,</span><span class="mi">9</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">  <span class="n">scatter!</span><span class="p">(</span><span class="n">plt1</span><span class="p">,[</span><span class="n">t</span><span class="p">,</span><span class="n">t</span><span class="p">],</span> <span class="n">sol3</span><span class="p">(</span><span class="n">t</span><span class="p">),</span> <span class="n">color</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="nb">false</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">plt</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">plt1</span><span class="p">,</span><span class="n">plt2</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">push!</span><span class="p">(</span><span class="n">list_plots</span><span class="p">,</span> <span class="n">plt</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">plot</span><span class="p">(</span><span class="n">list_plots</span><span class="p">[</span><span class="mi">100</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">anim</span> <span class="o">=</span> <span class="n">animate</span><span class="p">(</span><span class="n">list_plots</span><span class="p">,</span><span class="n">every</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+</span></span></code></pre></div>
 
 
 
@@ -916,18 +1109,16 @@ <h3 id="implicit-events">Implicit events</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/BB.gif" >
 
 
-  <img src="/img/BB.gif" alt=""  >
-</a>
 
 
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="BB.gif" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>The original curve is shown in black in the figure above.</p>
 <h2 id="sensitivity-analysis-with-events">Sensitivity analysis with events</h2>
@@ -942,7 +1133,7 @@ <h2 id="sensitivity-analysis-with-events">Sensitivity analysis with events</h2>
 <p>Inserting our results for $z(t) = z_{\rm exp}(t)$ instead of $z(t) = z_{\rm imp}(t)$ at $t_{\text{end}}$, a different value for the sensitivity is obtained (a value ignoring the changes in $z$ due to changes in the bouncing time $\tau$), e.g.,</p>
 <p>$$
 \quad \frac{\text{d}L^{\text{exp}}}{\text{d} z_0} = \begin{cases}
-1 &amp; \text{with } z(t) = z_{\rm exp}(t) ,\\<br>
+1 &amp; \text{with } z(t) = z_{\rm exp}(t) ,\\
 0.84 &amp; \text{with } z(t) = z_{\rm imp}(t) .
 \end{cases}
 $$</p>
@@ -953,7 +1144,7 @@ <h2 id="sensitivity-analysis-with-events">Sensitivity analysis with events</h2>
 <p>we could be interested in the sensitivity of $L$ with respect $g$. Using the velocity $v(t)$ at the explicit time $t = 0.9905$ instead of the implicit $v(\tau-)$, again a different value for the sensitivity is obtained (a value again ignoring the changes in $\tau$ and not the one we are looking for), even though $\tau = 0.9905$ in both cases:</p>
 <p>$$
 \quad \frac{\text{d}L}{\text{d} g} = \begin{cases}
--0.99005 &amp; \text{with } v(t) = v_{\rm exp}(t) ,\\<br>
+-0.99005 &amp; \text{with } v(t) = v_{\rm exp}(t) ,\\
 -\frac{g}{\sqrt{v_0^2 + 2 g z_0}} = -0.99995 &amp; \text{ with } v(t) = v_{\rm imp}(t) .
 \end{cases}
 $$</p>
@@ -970,51 +1161,48 @@ <h3 id="backsolve-adjoint-algorithm-for-ordinary-differential-equations">Backsol
 <p>We write</p>
 <p>$$\text{solve}(t_0, x_0, t, p)$$</p>
 <p>$(= x(t))$ for the functional solution of the ODE at time $t$.</p>
-<p>Regarding the computation of the sensitivities (the derivatives of the function <code>solve</code>), we may then choose one of the 
-<a href="https://diffeq.sciml.ai/stable/analysis/sensitivity/" target="_blank" rel="noopener">available algorithms</a> for the given differential equation. Currently, <code>BacksolveAdjoint()</code>, <code>InterpolatingAdjoint()</code>, <code>QuadratureAdjoint()</code>, <code>ReverseDiffAdjoint()</code>, <code>TrackerAdjoint()</code>, and <code>ForwardDiffAdjoint()</code> are compatible with events in ordinary differential equations.</p>
+<p>Regarding the computation of the sensitivities (the derivatives of the function <code>solve</code>), we may then choose one of the <a href="https://diffeq.sciml.ai/stable/analysis/sensitivity/" target="_blank" rel="noopener">available algorithms</a> for the given differential equation. Currently, <code>BacksolveAdjoint()</code>, <code>InterpolatingAdjoint()</code>, <code>QuadratureAdjoint()</code>, <code>ReverseDiffAdjoint()</code>, <code>TrackerAdjoint()</code>, and <code>ForwardDiffAdjoint()</code> are compatible with events in ordinary differential equations.</p>
 <p>Let us focus on the <code>BacksolveAdjoint()</code> algorithm which computes the sensitivities</p>
 <p>$$
 \begin{aligned}
-\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;= \lambda(t_{0}),\\<br>
+\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;= \lambda(t_{0}),\\
 \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}p} &amp;= \lambda_{p}(t_{0}),
 \end{aligned}
 $$</p>
 <p>of a loss function $L$ acting on the final state with respect to the initial state and the parameters. It does so by solving an ODE for $\lambda(s)$ in reverse time from $t$ to $t_0$</p>
 <p>$$
 \begin{aligned}
-\frac{\text{d}\lambda(s)}{\text{d}s} &amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\<br>
+\frac{\text{d}\lambda(s)}{\text{d}s} &amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\
 \frac{\text{d}\lambda_{p}(s)}{\text{d}s} &amp;= -\lambda(s)^\dagger \frac{\text{d} f(x(s), \rightarrow p, s)}{\text{d} p},
 \end{aligned}
 $$</p>
 <p>with initial conditions:</p>
 <p>$$
 \begin{aligned}
-\lambda(t)&amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\<br>
+\lambda(t)&amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\
 \lambda_{p}(t) &amp;= 0.
 \end{aligned}
 $$</p>
 <p>The arrows ($\rightarrow$) indicate the variable with respect to which we differentiate, which will become important later when the same variable shows up in multiple function arguments.</p>
-<p>Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In <code>BacksolveAdjoint()</code>, we recompute $x(s)$ &ndash; together with the adjoint variables &ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in 
-<a href="https://mitmath.github.io/18337/lecture11/adjoints" target="_blank" rel="noopener">Chris' MIT 18.337 lecture notes</a>.</p>
+<p>Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In <code>BacksolveAdjoint()</code>, we recompute $x(s)$ &ndash; together with the adjoint variables &ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in <a href="https://mitmath.github.io/18337/lecture11/adjoints" target="_blank" rel="noopener">Chris&rsquo; MIT 18.337 lecture notes</a>.</p>
 <p><code>BacksolveAdjoint()</code>, essentially the custom primitive differentiation rule of <code>solve</code>, is the elementary building block needed to derive sensitivities also in more complicated examples:</p>
 <p>Consider a loss depending on the state $x(s_i)$ at fix time points $s_i$ through loss functions $L_i^{\text{exp}}$,</p>
 <p>$$
 L^{\text{exp}} = \sum_i L_i^{\text{exp}}(s_i, x(s_i), p).
 $$</p>
 <p>Without contortions we can obtain the sensitivity of $L^{\text{exp}}$ in $p$ (or in $x_0$) using the tool we have. For those a bit familiar with automatic differentiation, this is perhaps easiest to see if we write $L^{\text{exp}}$ as pseudo code</p>
-<pre><code class="language-julia">function loss(t0, x0, p)
-    x1 = solve(t0, x0, s1, p)
-    L = L1(s1, x1, p)
-    x2 = solve(s1, x1, s2, p)
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-</code></pre>
-<p>and consider the problem of automatically differentiating it. You&rsquo;ll just need the primitives of <code>solve</code>, <code>L1</code> and <code>L2</code>! The sensitivities can be computed by repeated calls to <code>BacksolveAdjoint()</code> on the intervals <code>(s_i, s_{i+1})</code> backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation<sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> as single call <code>BacksolveAdjoint()</code> with discrete callbacks:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">loss</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">x1</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">s1</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">L</span> <span class="o">=</span> <span class="n">L1</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">x2</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">L</span> <span class="o">+=</span> <span class="n">L2</span><span class="p">(</span><span class="n">s2</span><span class="p">,</span> <span class="n">x2</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="o">...</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="n">L</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><p>and consider the problem of automatically differentiating it. You&rsquo;ll just need the primitives of <code>solve</code>, <code>L1</code> and <code>L2</code>! The sensitivities can be computed by repeated calls to <code>BacksolveAdjoint()</code> on the intervals <code>(s_i, s_{i+1})</code> backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation<sup id="fnref1:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> as single call <code>BacksolveAdjoint()</code> with discrete callbacks:</p>
 <p>$$
 \begin{aligned}
-\frac{\text{d}\lambda(t)}{\text{d}t} &amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\<br>
+\frac{\text{d}\lambda(t)}{\text{d}t} &amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\
 \frac{\text{d}\lambda_{p}(t)}{\text{d}t} &amp;= -\lambda(t)^\dagger \frac{\text{d} f(x(t), \rightarrow p, t)}{\text{d} p} - \frac{\text{d} L_i^{\text{exp}}(  x(t),\rightarrow p)}{\text{d} p}^\dagger \delta(t-s_i).
 \end{aligned}
 $$</p>
@@ -1028,24 +1216,23 @@ <h3 id="backsolve-adjoint-algorithm-for-ordinary-differential-equations">Backsol
 $$</p>
 <p>Now we can even properly define the <code>rrule</code> of <code>solve</code> in the sense of <code>DiffRules.jl</code>.</p>
 <h3 id="explicit-events-1">Explicit events</h3>
-<p>To make <code>BacksolveAdjoint()</code> compatible with explicit events<sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup>,</p>
-<pre><code class="language-julia">function loss(t0, x0, p)
-    x1- = solve(t0, x0, s1, p)
-    L = L1-(s1, x1-, p) # saved before affect
-    x1 = a(x1-, p, s1) # affect
-    L += L1(s1, x1, p) # saved after affect
-    x2- = solve(s1, x1, s2, p)
-    L += L2-(s2, x2-, p)
-    x2 = a(x2-, p, s2) # affect
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-</code></pre>
-<p>we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.<sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup> We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to</p>
+<p>To make <code>BacksolveAdjoint()</code> compatible with explicit events<sup id="fnref2:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup>,</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">loss</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">x1</span><span class="o">-</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">s1</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">L</span> <span class="o">=</span> <span class="n">L1</span><span class="o">-</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">x1</span><span class="o">-</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="c"># saved before affect</span>
+</span></span><span class="line"><span class="cl">    <span class="n">x1</span> <span class="o">=</span> <span class="n">a</span><span class="p">(</span><span class="n">x1</span><span class="o">-</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">s1</span><span class="p">)</span> <span class="c"># affect</span>
+</span></span><span class="line"><span class="cl">    <span class="n">L</span> <span class="o">+=</span> <span class="n">L1</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span> <span class="c"># saved after affect</span>
+</span></span><span class="line"><span class="cl">    <span class="n">x2</span><span class="o">-</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">L</span> <span class="o">+=</span> <span class="n">L2</span><span class="o">-</span><span class="p">(</span><span class="n">s2</span><span class="p">,</span> <span class="n">x2</span><span class="o">-</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">x2</span> <span class="o">=</span> <span class="n">a</span><span class="p">(</span><span class="n">x2</span><span class="o">-</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">s2</span><span class="p">)</span> <span class="c"># affect</span>
+</span></span><span class="line"><span class="cl">    <span class="n">L</span> <span class="o">+=</span> <span class="n">L2</span><span class="p">(</span><span class="n">s2</span><span class="p">,</span> <span class="n">x2</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="o">...</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="n">L</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><p>we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.<sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup> We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to</p>
 <p>$$
 \begin{aligned}
-\lambda({s_j}-) &amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\<br>
+\lambda({s_j}-) &amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\
 \lambda_p({s_j}-) &amp;= \lambda_p({s_j}) -  \lambda({s_j})^\dagger \frac{\text{d} a(x({s_j}-), \rightarrow p, {s_j}-)}{\text{d} p}
 \end{aligned}
 $$</p>
@@ -1059,27 +1246,26 @@ <h3 id="implicit-events-1">Implicit events</h3>
 <p>to put emphasis on this, or equivalently, compute for a unspecified function $L(t, x, p)$ the result of $\frac{\text{d}L}{\text{d}p}$ with</p>
 <p>$$
 \begin{aligned}
-\frac{\text{d}L}{\text{d}p} &amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\<br>
+\frac{\text{d}L}{\text{d}p} &amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\
 &amp;= \frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p},
 \end{aligned}
 $$</p>
 <p>which indicates that changing $p$ influences $L$ both through changes in $\tau_1$ as well as changes in</p>
 <p>$$\xi_1 = x(\tau_1-).$$</p>
 <p>This case where we have a loss function $L = L_1$ depending on $\tau_1$, $x(\tau_1)$, and $p$
-was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>.</p>
+was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper<sup id="fnref1:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>.</p>
 <p>Therefore, the sensitivity of the event time with respect to parameters $\frac{\text{d}\tau}{\text{d}p}$ must be taken into account.
-Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>.</p>
-<p>In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the 
-<a href="https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf" target="_blank" rel="noopener">implicit function theorem</a>. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields</p>
+Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way<sup id="fnref2:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>.</p>
+<p>In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the <a href="https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf" target="_blank" rel="noopener">implicit function theorem</a>. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields</p>
 <p>$$
 \begin{aligned}
-\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\<br>
+\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\
 \end{aligned}
 $$</p>
 <p>The total derivative<sup id="fnref:6"><a href="#fn:6" class="footnote-ref" role="doc-noteref">6</a></sup> inside the bracket is:
 $$
 \begin{aligned}
-\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\<br>
+\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\
 \end{aligned}
 $$</p>
 <p>Since</p>
@@ -1102,8 +1288,8 @@ <h3 id="implicit-events-1">Implicit events</h3>
 <p>We can now write the gradient as:</p>
 <p>$$
 \begin{aligned}
-\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\<br>
-+&amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\<br>
+\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\
++&amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\
 +&amp; \frac{\text{d}L(\rightarrow \tau_1(p), \text{solve}(t_0, x_0, \rightarrow \tau_1(p),  p), p)}{\text{d}\tau_1} \frac{\text{d} \tau_1(p)}{\text{d}p},
 \end{aligned}
 $$</p>
@@ -1172,7 +1358,7 @@ <h3 id="implicit-events-1">Implicit events</h3>
 <li>If both limits contribute to the loss function, the contributions ${\color{red}\lambda_1}$ and ${\color{blue}\lambda_{a,1}}$ are added.</li>
 </ol>
 <h4 id="generalization-several-events">Generalization: several events</h4>
-<p>As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle<sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:</p>
+<p>As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle<sup id="fnref1:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:</p>
 <p>$$
 \text{solve}(t_0, x_0, t, p)  = \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p),
 $$</p>
@@ -1180,16 +1366,15 @@ <h4 id="generalization-several-events">Generalization: several events</h4>
 <p>(A good exercise to get familiar with these type of arguments is to verify</p>
 <p>$$\frac{\text{d}}{\text{d}s} \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p) = 0.)$$</p>
 <p>Essentially, we know how to address several events already by considering a loss function</p>
-<pre><code class="language-julia">function loss2(t0, x0, p)
-    tau1, xi1 = solve2(t0, x0, g, p)
-    L = L1(tau1, xi1, p)
-    tau2, xi2 = solve2(tau1, xi1, g, p)
-    L += L2(tau2, xi2, p)
-    ...
-    return L
-end
-</code></pre>
-<p>and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for <code>solve2</code> (<code>rrule</code> for <code>solve2</code>) we derived above.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">loss2</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">tau1</span><span class="p">,</span> <span class="n">xi1</span> <span class="o">=</span> <span class="n">solve2</span><span class="p">(</span><span class="n">t0</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">L</span> <span class="o">=</span> <span class="n">L1</span><span class="p">(</span><span class="n">tau1</span><span class="p">,</span> <span class="n">xi1</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">tau2</span><span class="p">,</span> <span class="n">xi2</span> <span class="o">=</span> <span class="n">solve2</span><span class="p">(</span><span class="n">tau1</span><span class="p">,</span> <span class="n">xi1</span><span class="p">,</span> <span class="n">g</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">L</span> <span class="o">+=</span> <span class="n">L2</span><span class="p">(</span><span class="n">tau2</span><span class="p">,</span> <span class="n">xi2</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="o">...</span>
+</span></span><span class="line"><span class="cl">    <span class="k">return</span> <span class="n">L</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><p>and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for <code>solve2</code> (<code>rrule</code> for <code>solve2</code>) we derived above.</p>
 <p>Differentiating by hand, we have the following modification of the method from the previous section, which can be derived choosing a particular function $L$ in the previous section which incorporates the <code>solve</code> from $\tau_1$ to $t_{\text{end}}$. Note</p>
 <p>$$
 \lambda_{\tau_1}^0 = \frac{\text{d}(\text{solve}(\rightarrow \tau_1,  a(\xi_1,p), t_{\text{end}}, p))}{\text{d} \tau_1} = - f(a(\xi_1,p), p, \tau_1).
@@ -1221,43 +1406,38 @@ <h4 id="generalization-several-events">Generalization: several events</h4>
 <li>Backpropagate $\lambda_J = {\color{red}\lambda_J} + {\color{blue}\lambda_{a,J}} + {\color{green}\lambda_{c,J}}$ to the next event time $\tau_{J-1}$ and iterate over the remaining $J-1$ events.</li>
 </ol>
 <h2 id="outlook">Outlook</h2>
-<p>We are still refining the adjoints in case of implicit discontinuities (<code>ContinuousCallbacks</code>). For further information, the interested reader is encouraged to track the associated issues 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/issues/383" target="_blank" rel="noopener">#383</a> and 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/issues/374" target="_blank" rel="noopener">#374</a>, and 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/445" target="_blank" rel="noopener">PR #445</a> in the DiffEqSensitivity.jl package.</p>
-<p>If you have any questions or comments, please don’t hesitate to contact us (
-<a href="https://github.com/frankschae" target="_blank" rel="noopener">github.com/frankschae</a>)!</p>
-<section class="footnotes" role="doc-endnotes">
+<p>We are still refining the adjoints in case of implicit discontinuities (<code>ContinuousCallbacks</code>). For further information, the interested reader is encouraged to track the associated issues <a href="https://github.com/SciML/DiffEqSensitivity.jl/issues/383" target="_blank" rel="noopener">#383</a> and <a href="https://github.com/SciML/DiffEqSensitivity.jl/issues/374" target="_blank" rel="noopener">#374</a>, and <a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/445" target="_blank" rel="noopener">PR #445</a> in the DiffEqSensitivity.jl package.</p>
+<p>If you have any questions or comments, please don’t hesitate to contact us (<a href="https://github.com/frankschae" target="_blank" rel="noopener">github.com/frankschae</a>)!</p>
+<div class="footnotes" role="doc-endnotes">
 <hr>
 <ol>
-<li id="fn:1" role="doc-endnote">
-<p>Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). <a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:1">
+<p>Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&#160;<a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:2" role="doc-endnote">
-<p>Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). <a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:2">
+<p>Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&#160;<a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref2:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:3" role="doc-endnote">
-<p>Timo C. Wunderlich and Christian Pehle, Sci. Rep. <em>11</em>, 12829 (2021). <a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:3">
+<p>Timo C. Wunderlich and Christian Pehle, Sci. Rep. <em>11</em>, 12829 (2021).&#160;<a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:4" role="doc-endnote">
-<p>Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). <a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:4">
+<p>Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&#160;<a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref2:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:5" role="doc-endnote">
-<p>If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$. <a href="#fnref:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:5">
+<p>If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$.&#160;<a href="#fnref:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:6" role="doc-endnote">
-<p>For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the 
-<a href="https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables" target="_blank" rel="noopener">total derivative</a> with respect to the independent variable $t$ is given by the sum of all partial derivatives
+<li id="fn:6">
+<p>For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the <a href="https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables" target="_blank" rel="noopener">total derivative</a> with respect to the independent variable $t$ is given by the sum of all partial derivatives
 $$
 \begin{aligned}
-\frac{\text{d}y}{\text{d}t} &amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\<br>
-&amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\<br>
+\frac{\text{d}y}{\text{d}t} &amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\
+&amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\
 &amp;+ \frac{\text{d}f(t, x_1(t),x_2(\rightarrow t),\dots,x_N(t))}{\text{d}t} + \dots +  \frac{\text{d}f(t, x_1(t),x_2(t),\dots,x_N(\rightarrow t))}{\text{d}t}.
 \end{aligned}
-$$ <a href="#fnref:6" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+$$&#160;<a href="#fnref:6" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
 </ol>
-</section>
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-<p>In this project, we have implemented new promising tools within the 
-<a href="https://sciml.ai" target="_blank" rel="noopener">SciML</a> organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
+<p>In this project, we have implemented new promising tools within the <a href="https://sciml.ai" target="_blank" rel="noopener">SciML</a> organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
 The high weak order solvers will allow for massive performance advantages for fitting expectations of equations.
 Instead of automatic differentiation (AD) through the operations of an SDE solver, which scales poorly in memory, one can now use efficient stochastic adjoint sensitivity methods.</p>
 <h2 id="blog-posts">Blog posts</h2>
 <p>The following posts describe the work during the entire period in more detail:</p>
 <ol>
-<li>
-<a href="https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" target="_blank" rel="noopener">GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a></li>
-<li>
-<a href="https://frankschae.github.io/post/high-weak/" target="_blank" rel="noopener">High weak order SDE solvers</a></li>
+<li><a href="https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" target="_blank" rel="noopener">GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a></li>
+<li><a href="https://frankschae.github.io/post/high-weak/" target="_blank" rel="noopener">High weak order SDE solvers</a></li>
 </ol>
 <h2 id="docs">Docs</h2>
-<p>The documentation of the solvers is available 
-<a href="https://diffeq.sciml.ai/latest/solvers/sde_solve/#High-Weak-Order-Methods" target="_blank" rel="noopener">here</a>.
-Docs with respect to the adjoint sensitivity tools will be available 
-<a href="https://diffeq.sciml.ai/latest/analysis/sensitivity/" target="_blank" rel="noopener">here</a>.</p>
+<p>The documentation of the solvers is available <a href="https://diffeq.sciml.ai/latest/solvers/sde_solve/#High-Weak-Order-Methods" target="_blank" rel="noopener">here</a>.
+Docs with respect to the adjoint sensitivity tools will be available <a href="https://diffeq.sciml.ai/latest/analysis/sensitivity/" target="_blank" rel="noopener">here</a>.</p>
 <h2 id="achievements">Achievements</h2>
 <p>Please find below a list of the PRs carried out during GSoC in the different repositories in chronological order within the SciML ecosystem.</p>
 <h4 id="stochasticdiffeqjl">StochasticDiffEq.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/285" target="_blank" rel="noopener">Inplace version of DRI1 scheme</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/289" target="_blank" rel="noopener">RI1 method</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/305" target="_blank" rel="noopener">tstop fixes for reverse time propagation</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/317" target="_blank" rel="noopener">Fixes for EulerHeun scheme</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/327" target="_blank" rel="noopener">Speed up the tests for the DRI1 and RI1 schemes</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/328" target="_blank" rel="noopener">RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/329" target="_blank" rel="noopener">RDI1WM scheme</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/332" target="_blank" rel="noopener">Adaptive version of the stochastic Runge-Kutta schemes by embedding</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/333" target="_blank" rel="noopener">RS1 and RS2 schemes</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/334" target="_blank" rel="noopener">PL1WM scheme</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/337" target="_blank" rel="noopener">NON scheme</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/338" target="_blank" rel="noopener">Citations for weak methods</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/342" target="_blank" rel="noopener">Stochastic improved and modified Euler methods</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/343" target="_blank" rel="noopener">COM scheme</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/348" target="_blank" rel="noopener">Computationally more efficient NON variant (NON2)</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/285" target="_blank" rel="noopener">Inplace version of DRI1 scheme</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/289" target="_blank" rel="noopener">RI1 method</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/305" target="_blank" rel="noopener">tstop fixes for reverse time propagation</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/317" target="_blank" rel="noopener">Fixes for EulerHeun scheme</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/327" target="_blank" rel="noopener">Speed up the tests for the DRI1 and RI1 schemes</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/328" target="_blank" rel="noopener">RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/329" target="_blank" rel="noopener">RDI1WM scheme</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/332" target="_blank" rel="noopener">Adaptive version of the stochastic Runge-Kutta schemes by embedding</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/333" target="_blank" rel="noopener">RS1 and RS2 schemes</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/334" target="_blank" rel="noopener">PL1WM scheme</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/337" target="_blank" rel="noopener">NON scheme</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/338" target="_blank" rel="noopener">Citations for weak methods</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/342" target="_blank" rel="noopener">Stochastic improved and modified Euler methods</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/343" target="_blank" rel="noopener">COM scheme</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/348" target="_blank" rel="noopener">Computationally more efficient NON variant (NON2)</a></li>
 </ul>
 <p>Open:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/288" target="_blank" rel="noopener">Static array tests</a></li>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/347" target="_blank" rel="noopener">Levy area for non-commutative noise processes</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/288" target="_blank" rel="noopener">Static array tests</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/347" target="_blank" rel="noopener">Levy area for non-commutative noise processes</a></li>
 </ul>
 <h4 id="diffeqsensitivityjl">DiffEqSensitivity.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/235" target="_blank" rel="noopener">Adjoint sensitivities for steady states</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/237" target="_blank" rel="noopener">Concrete_solve dispatch for steady state problem</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/242" target="_blank" rel="noopener">BacksolveAdjoint for SDEs</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/256" target="_blank" rel="noopener">GPU savety for SDE BacksolveAdjoint</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/258" target="_blank" rel="noopener">Tests for concrete solve with respect to SDEs</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/260" target="_blank" rel="noopener">Alternative differentiation choices (vjps) for noise Jacobian</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/265" target="_blank" rel="noopener">Fixes and tests for inplace formulation of BacksolveAdjoint</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/268" target="_blank" rel="noopener">Efficient SDE BacksolveAdjoint for scalar noise</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/275" target="_blank" rel="noopener">Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/295" target="_blank" rel="noopener">InterpolatingAdjoint for SDEs</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/298" target="_blank" rel="noopener">Citations for backsolve, steadystate and interpolation adjoint</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/299" target="_blank" rel="noopener">Allow for more general noise processes: replace NoiseGrid by NoiseWrapper</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/303" target="_blank" rel="noopener">Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/305" target="_blank" rel="noopener">Cheaper non-diagonal noise tests</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/235" target="_blank" rel="noopener">Adjoint sensitivities for steady states</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/237" target="_blank" rel="noopener">Concrete_solve dispatch for steady state problem</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/242" target="_blank" rel="noopener">BacksolveAdjoint for SDEs</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/256" target="_blank" rel="noopener">GPU savety for SDE BacksolveAdjoint</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/258" target="_blank" rel="noopener">Tests for concrete solve with respect to SDEs</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/260" target="_blank" rel="noopener">Alternative differentiation choices (vjps) for noise Jacobian</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/265" target="_blank" rel="noopener">Fixes and tests for inplace formulation of BacksolveAdjoint</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/268" target="_blank" rel="noopener">Efficient SDE BacksolveAdjoint for scalar noise</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/275" target="_blank" rel="noopener">Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/295" target="_blank" rel="noopener">InterpolatingAdjoint for SDEs</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/298" target="_blank" rel="noopener">Citations for backsolve, steadystate and interpolation adjoint</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/299" target="_blank" rel="noopener">Allow for more general noise processes: replace NoiseGrid by NoiseWrapper</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/303" target="_blank" rel="noopener">Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/305" target="_blank" rel="noopener">Cheaper non-diagonal noise tests</a></li>
 </ul>
 <p>Open:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/317" target="_blank" rel="noopener">Support adjoints for SDEs written in the Ito sense</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/317" target="_blank" rel="noopener">Support adjoints for SDEs written in the Ito sense</a></li>
 </ul>
 <h4 id="diffeqnoiseprocessjl">DiffEqNoiseProcess.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/48" target="_blank" rel="noopener">Multi-dimensional Brownian motion tests</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/49" target="_blank" rel="noopener">Bug fix for inplace form of NoiseGrid</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/51" target="_blank" rel="noopener">Reversible NoiseWrapper</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/53" target="_blank" rel="noopener">Relax the size constraints of the available noise processes</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/54" target="_blank" rel="noopener">Generalization of the real-valued white noise process function</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/55" target="_blank" rel="noopener">Fix of an extraction issue with NoiseGrid</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56" target="_blank" rel="noopener">Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56" target="_blank" rel="noopener">Extraction and endpoint fixes on NoiseWrapper</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/62" target="_blank" rel="noopener">Reversal of SDEs written in the Ito sense</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/48" target="_blank" rel="noopener">Multi-dimensional Brownian motion tests</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/49" target="_blank" rel="noopener">Bug fix for inplace form of NoiseGrid</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/51" target="_blank" rel="noopener">Reversible NoiseWrapper</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/53" target="_blank" rel="noopener">Relax the size constraints of the available noise processes</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/54" target="_blank" rel="noopener">Generalization of the real-valued white noise process function</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/55" target="_blank" rel="noopener">Fix of an extraction issue with NoiseGrid</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56" target="_blank" rel="noopener">Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56" target="_blank" rel="noopener">Extraction and endpoint fixes on NoiseWrapper</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/62" target="_blank" rel="noopener">Reversal of SDEs written in the Ito sense</a></li>
 </ul>
 <h4 id="diffeqgpujl">DiffEqGPU.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqGPU.jl/pull/59" target="_blank" rel="noopener">Memory efficient reduction function for ensemble problems</a></li>
+<li><a href="https://github.com/SciML/DiffEqGPU.jl/pull/59" target="_blank" rel="noopener">Memory efficient reduction function for ensemble problems</a></li>
 </ul>
 <h4 id="modelingtoolkitjl">ModelingToolkit.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/317" target="_blank" rel="noopener">modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/317" target="_blank" rel="noopener">modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense</a></li>
 </ul>
 <h4 id="diffeqdevtoolsjl">DiffEqDevTools.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqDevTools.jl/pull/62" target="_blank" rel="noopener">NoiseWrapper alternative for analyticless convergence tests of SDE solvers</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqDevTools.jl/pull/74" target="_blank" rel="noopener">test_convergence() dispatch for ensemble simulations</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqDevTools.jl/pull/75" target="_blank" rel="noopener">Work precision set for ensemble problems</a></li>
+<li><a href="https://github.com/SciML/DiffEqDevTools.jl/pull/62" target="_blank" rel="noopener">NoiseWrapper alternative for analyticless convergence tests of SDE solvers</a></li>
+<li><a href="https://github.com/SciML/DiffEqDevTools.jl/pull/74" target="_blank" rel="noopener">test_convergence() dispatch for ensemble simulations</a></li>
+<li><a href="https://github.com/SciML/DiffEqDevTools.jl/pull/75" target="_blank" rel="noopener">Work precision set for ensemble problems</a></li>
 </ul>
 <h4 id="diffeqbasejl">DiffEqBase.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqBase.jl/pull/503" target="_blank" rel="noopener">Fix concrete_solve tests</a></li>
+<li><a href="https://github.com/SciML/DiffEqBase.jl/pull/503" target="_blank" rel="noopener">Fix concrete_solve tests</a></li>
 </ul>
 <h2 id="future-work">Future work</h2>
 <p>There is still a lot that we&rsquo;d like to do, e.g.,</p>
@@ -814,12 +961,8 @@ <h2 id="future-work">Future work</h2>
 </ul>
 <p>Contributions, suggestions &amp; comments are always welcome! You might like to join our slac channels #diffeq-bridged and #neuralsde to get in touch.</p>
 <h2 id="acknowledgement">Acknowledgement</h2>
-<p>I would like to thank my mentors 
-<a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, 
-<a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, and  
-<a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a> for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback.
-I would also like to thank 
-<a href="https://www.quantumtheory-bruder.physik.unibas.ch/people.html" target="_blank" rel="noopener">Christoph Bruder</a>, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts.
+<p>I would like to thank my mentors <a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, <a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, and  <a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a> for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback.
+I would also like to thank <a href="https://www.quantumtheory-bruder.physik.unibas.ch/people.html" target="_blank" rel="noopener">Christoph Bruder</a>, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts.
 Many thanks to the very supportive julia community and to Google&rsquo;s open source program for funding this experience!</p>
 
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+        <a href="https://twitter.com/intent/tweet?url=https%3A%2F%2Ffrankschae.github.io%2Fpost%2Fgsoc-2020%2F&amp;text=High&#43;weak&#43;order&#43;solvers&#43;and&#43;adjoint&#43;sensitivity&#43;analysis&#43;for&#43;stochastic&#43;differential&#43;equations" target="_blank" rel="noopener" class="share-btn-twitter" aria-label="twitter">
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+        <a href="https://www.facebook.com/sharer.php?u=https%3A%2F%2Ffrankschae.github.io%2Fpost%2Fgsoc-2020%2F&amp;t=High&#43;weak&#43;order&#43;solvers&#43;and&#43;adjoint&#43;sensitivity&#43;analysis&#43;for&#43;stochastic&#43;differential&#43;equations" target="_blank" rel="noopener" class="share-btn-facebook" aria-label="facebook">
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+        <a href="mailto:?subject=High%20weak%20order%20solvers%20and%20adjoint%20sensitivity%20analysis%20for%20stochastic%20differential%20equations&amp;body=https%3A%2F%2Ffrankschae.github.io%2Fpost%2Fgsoc-2020%2F" target="_blank" rel="noopener" class="share-btn-email" aria-label="envelope">
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-        <a href="https://web.whatsapp.com/send?text=High%20weak%20order%20solvers%20and%20adjoint%20sensitivity%20analysis%20for%20stochastic%20differential%20equations%20https://frankschae.github.io/post/gsoc-2020/" target="_blank" rel="noopener" class="share-btn-whatsapp">
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+      
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+      
+      
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-        <h5 class="card-title"><a href="https://frankschae.github.io/">Frank Schäfer</a></h5>
-        <h6 class="card-subtitle">Postdoctoral researcher</h6>
-        <p class="card-text">My research interests include many-body physics, machine learning, and differentiable programming.</p>
-        <ul class="network-icon" aria-hidden="true">
+    <div class="media-body">
+      <h5 class="card-title"><a href="https://frankschae.github.io/">Frank Schäfer</a></h5>
+      <h6 class="card-subtitle">Postdoctoral researcher</h6>
+      <p class="card-text">My research interests include many-body physics, probabilistic machine learning, and differentiable programming.</p>
+      <ul class="network-icon" aria-hidden="true">
   
     
     
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@@ -656,128 +855,91 @@ <h1>Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis</h1>
     <div class="article-style">
       <h2 id="project-summary">Project summary</h2>
 <p>In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.</p>
-<p>Possible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available 
-<a href="https://summerofcode.withgoogle.com/projects/#5357798591823872" target="_blank" rel="noopener">on the GSoC page</a>.</p>
+<p>Possible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available <a href="https://summerofcode.withgoogle.com/projects/#5357798591823872" target="_blank" rel="noopener">on the GSoC page</a>.</p>
 <h2 id="blog-posts">Blog posts</h2>
 <p>The following blog posts describe the work throughout the GSoC period in more detail:</p>
 <ol>
-<li>
-<a href="https://frankschae.github.io/post/hybridde/" target="_blank" rel="noopener">Neural Hybrid Differential Equations</a></li>
-<li>
-<a href="https://frankschae.github.io/post/shadowing/" target="_blank" rel="noopener">Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></li>
-<li>
-<a href="https://frankschae.github.io/post/bouncing_ball/" target="_blank" rel="noopener">Sensitivity Analysis of Hybrid Differential Equations</a></li>
-<li>
-<a href="https://frankschae.github.io/post/abstract_differentiation/" target="_blank" rel="noopener">AbstractDifferentiation.jl for AD-backend agnostic code</a></li>
+<li><a href="https://frankschae.github.io/post/hybridde/" target="_blank" rel="noopener">Neural Hybrid Differential Equations</a></li>
+<li><a href="https://frankschae.github.io/post/shadowing/" target="_blank" rel="noopener">Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></li>
+<li><a href="https://frankschae.github.io/post/bouncing_ball/" target="_blank" rel="noopener">Sensitivity Analysis of Hybrid Differential Equations</a></li>
+<li><a href="https://frankschae.github.io/post/abstract_differentiation/" target="_blank" rel="noopener">AbstractDifferentiation.jl for AD-backend agnostic code</a></li>
 </ol>
 <h2 id="docs">Docs</h2>
-<p>Documentation with respect to the adjoint sensitivity tools will be available 
-<a href="https://diffeq.sciml.ai/latest/analysis/sensitivity/" target="_blank" rel="noopener">on the local sensitivity analysis</a> and 
-<a href="http://scimlbase.sciml.ai/dev/fundamentals/Differentiation/" target="_blank" rel="noopener">on the control of automatic differentiation choices</a> pages.</p>
+<p>Documentation with respect to the adjoint sensitivity tools will be available <a href="https://diffeq.sciml.ai/latest/analysis/sensitivity/" target="_blank" rel="noopener">on the local sensitivity analysis</a> and <a href="http://scimlbase.sciml.ai/dev/fundamentals/Differentiation/" target="_blank" rel="noopener">on the control of automatic differentiation choices</a> pages.</p>
 <h2 id="achievements">Achievements</h2>
 <p>Below is a list of PRs in the various repositories in chronological order.</p>
 <h4 id="diffeqsensitivityjl">DiffEqSensitivity.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/415" target="_blank" rel="noopener">Add additive noise downstream test for DiffEqFlux</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/416" target="_blank" rel="noopener">DiscreteCallback fixes</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/417" target="_blank" rel="noopener">Allow for changes of p in callbacks</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/418" target="_blank" rel="noopener">Fix for using the correct uleft/pleft in continuous callback</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/419" target="_blank" rel="noopener">Fix broadcasting error on steady state adjoint</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/420" target="_blank" rel="noopener">Forward Least Squares Shadowing (LSS)</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/422" target="_blank" rel="noopener">Adjoint-mode for the LSS method</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/423" target="_blank" rel="noopener">concrete_solve dispatch for LSS methods</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/437" target="_blank" rel="noopener">Non-Intrusive Least Square Shadowing (NILSS)</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/442" target="_blank" rel="noopener">concrete_solve for NILSS</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/443" target="_blank" rel="noopener">Remove allocation in NILSS</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/444" target="_blank" rel="noopener">Handle additional callback case</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/445" target="_blank" rel="noopener">State-dependent Continuous Callbacks for BacksolveAdjoint</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/474" target="_blank" rel="noopener">QuadratureAdjoint() for ContinuousCallback</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/475" target="_blank" rel="noopener">More tests for Neural ODEs with callbacks for different sensitivity algorithms</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/476" target="_blank" rel="noopener">Support for PeriodicCallbacks in continuous adjoint methods</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/415" target="_blank" rel="noopener">Add additive noise downstream test for DiffEqFlux</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/416" target="_blank" rel="noopener">DiscreteCallback fixes</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/417" target="_blank" rel="noopener">Allow for changes of p in callbacks</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/418" target="_blank" rel="noopener">Fix for using the correct uleft/pleft in continuous callback</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/419" target="_blank" rel="noopener">Fix broadcasting error on steady state adjoint</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/420" target="_blank" rel="noopener">Forward Least Squares Shadowing (LSS)</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/422" target="_blank" rel="noopener">Adjoint-mode for the LSS method</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/423" target="_blank" rel="noopener">concrete_solve dispatch for LSS methods</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/437" target="_blank" rel="noopener">Non-Intrusive Least Square Shadowing (NILSS)</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/442" target="_blank" rel="noopener">concrete_solve for NILSS</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/443" target="_blank" rel="noopener">Remove allocation in NILSS</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/444" target="_blank" rel="noopener">Handle additional callback case</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/445" target="_blank" rel="noopener">State-dependent Continuous Callbacks for BacksolveAdjoint</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/474" target="_blank" rel="noopener">QuadratureAdjoint() for ContinuousCallback</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/475" target="_blank" rel="noopener">More tests for Neural ODEs with callbacks for different sensitivity algorithms</a></li>
+<li><a href="https://github.com/SciML/DiffEqSensitivity.jl/pull/476" target="_blank" rel="noopener">Support for PeriodicCallbacks in continuous adjoint methods</a></li>
 </ul>
 <h4 id="abstractdifferentiationjl">AbstractDifferentiation.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/2" target="_blank" rel="noopener">Fixes gradient, Jacobian, Hessian, and vjp tests</a></li>
+<li><a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/2" target="_blank" rel="noopener">Fixes gradient, Jacobian, Hessian, and vjp tests</a></li>
 </ul>
 <p>Open:</p>
 <ul>
-<li>
-<a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/3" target="_blank" rel="noopener">Add ForwardDiff and Zygote</a></li>
+<li><a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/3" target="_blank" rel="noopener">Add ForwardDiff and Zygote</a></li>
 </ul>
 <h4 id="ordinarydiffeqjl">OrdinaryDiffEq.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/OrdinaryDiffEq.jl/pull/1424" target="_blank" rel="noopener">Fix discrete reverse mode for some standard controllers</a></li>
+<li><a href="https://github.com/SciML/OrdinaryDiffEq.jl/pull/1424" target="_blank" rel="noopener">Fix discrete reverse mode for some standard controllers</a></li>
 </ul>
 <h4 id="diffeqcallbacksjl">DiffEqCallbacks.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqCallbacks.jl/pull/102" target="_blank" rel="noopener">Introduce a PeriodicCallbackAffect struct</a></li>
+<li><a href="https://github.com/SciML/DiffEqCallbacks.jl/pull/102" target="_blank" rel="noopener">Introduce a PeriodicCallbackAffect struct</a></li>
 </ul>
 <h4 id="steadystatediffeqjl">SteadyStateDiffEq.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/SteadyStateDiffEq.jl/pull/31" target="_blank" rel="noopener">Convert alg.tspan to type of prob.u0</a></li>
+<li><a href="https://github.com/SciML/SteadyStateDiffEq.jl/pull/31" target="_blank" rel="noopener">Convert alg.tspan to type of prob.u0</a></li>
 </ul>
 <h4 id="chainrulesjl">ChainRules.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/JuliaDiff/ChainRules.jl/pull/506" target="_blank" rel="noopener">Do not differentiate through the construction of BitArray</a></li>
-<li>
-<a href="https://github.com/JuliaDiff/ChainRules.jl/pull/508" target="_blank" rel="noopener">Use splatting in BitArray</a></li>
+<li><a href="https://github.com/JuliaDiff/ChainRules.jl/pull/506" target="_blank" rel="noopener">Do not differentiate through the construction of BitArray</a></li>
+<li><a href="https://github.com/JuliaDiff/ChainRules.jl/pull/508" target="_blank" rel="noopener">Use splatting in BitArray</a></li>
 </ul>
 <h4 id="diffeqnoiseprocessjl">DiffEqNoiseProcess.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/94" target="_blank" rel="noopener">Allow solvers to use Noise Grid with SVectors</a></li>
+<li><a href="https://github.com/SciML/DiffEqNoiseProcess.jl/pull/94" target="_blank" rel="noopener">Allow solvers to use Noise Grid with SVectors</a></li>
 </ul>
 <h4 id="stochasticdiffeqjl">StochasticDiffEq.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/StochasticDiffEq.jl/pull/428" target="_blank" rel="noopener">Remove Ihat2 matrix from weak solvers</a></li>
+<li><a href="https://github.com/SciML/StochasticDiffEq.jl/pull/428" target="_blank" rel="noopener">Remove Ihat2 matrix from weak solvers</a></li>
 </ul>
 <h4 id="diffeqdocsjl">DiffEqDocs.jl</h4>
 <p>Merged:</p>
 <ul>
-<li>
-<a href="https://github.com/SciML/DiffEqDocs.jl/pull/490" target="_blank" rel="noopener">Small typo on plot page</a></li>
-<li>
-<a href="https://github.com/SciML/DiffEqDocs.jl/pull/492" target="_blank" rel="noopener">Add docs for shadowing methods</a></li>
+<li><a href="https://github.com/SciML/DiffEqDocs.jl/pull/490" target="_blank" rel="noopener">Small typo on plot page</a></li>
+<li><a href="https://github.com/SciML/DiffEqDocs.jl/pull/492" target="_blank" rel="noopener">Add docs for shadowing methods</a></li>
 </ul>
 <h2 id="future-work">Future work</h2>
 <p>Besides the implementation of more shadowing methods, such as</p>
 <ul>
-<li>
-<a href="https://arxiv.org/abs/1801.08674" target="_blank" rel="noopener">NILSAS</a>,</li>
-<li>
-<a href="https://arxiv.org/abs/1711.06633" target="_blank" rel="noopener">FD-NILSS</a>, or</li>
-<li>
-<a href="https://arxiv.org/abs/2009.00595" target="_blank" rel="noopener">Fast linear response</a>,</li>
+<li><a href="https://arxiv.org/abs/1801.08674" target="_blank" rel="noopener">NILSAS</a>,</li>
+<li><a href="https://arxiv.org/abs/1711.06633" target="_blank" rel="noopener">FD-NILSS</a>, or</li>
+<li><a href="https://arxiv.org/abs/2009.00595" target="_blank" rel="noopener">Fast linear response</a>,</li>
 </ul>
 <p>we are planning to</p>
 <ul>
@@ -785,20 +947,10 @@ <h2 id="future-work">Future work</h2>
 <li>refine the AbstractDifferentiation.jl package and use it within DiffEqSensitivity.jl,</li>
 <li>add more docs and examples.</li>
 </ul>
-<p>If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the 
-<a href="https://discourse.julialang.org/" target="_blank" rel="noopener">Julia language discourse</a>.</p>
+<p>If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the <a href="https://discourse.julialang.org/" target="_blank" rel="noopener">Julia language discourse</a>.</p>
 <h2 id="acknowledgement">Acknowledgement</h2>
-<p>Many thanks to my mentors 
-<a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, 
-<a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, 
-<a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a>, and 
-<a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a> for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times.
-I would also like to thank 
-<a href="https://quantumtheory-bruder.physik.unibas.ch/en/people/group-members/christoph-bruder/" target="_blank" rel="noopener">Christoph Bruder</a>, 
-<a href="https://github.com/arnoldjulian" target="_blank" rel="noopener">Julian Arnold</a>, and 
-<a href="https://github.com/mako-git" target="_blank" rel="noopener">Martin Koppenhöfer</a> for helpful comments on my blog posts. Special thanks to 
-<a href="https://github.com/Zymrael" target="_blank" rel="noopener">Michael Poli</a> and 
-<a href="https://github.com/massastrello" target="_blank" rel="noopener">Stefano Massaroli</a> for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google&rsquo;s open source program for funding this experience!</p>
+<p>Many thanks to my mentors <a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, <a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, <a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a>, and <a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a> for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times.
+I would also like to thank <a href="https://quantumtheory-bruder.physik.unibas.ch/en/people/group-members/christoph-bruder/" target="_blank" rel="noopener">Christoph Bruder</a>, <a href="https://github.com/arnoldjulian" target="_blank" rel="noopener">Julian Arnold</a>, and <a href="https://github.com/mako-git" target="_blank" rel="noopener">Martin Koppenhöfer</a> for helpful comments on my blog posts. Special thanks to <a href="https://github.com/Zymrael" target="_blank" rel="noopener">Michael Poli</a> and <a href="https://github.com/massastrello" target="_blank" rel="noopener">Stefano Massaroli</a> for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google&rsquo;s open source program for funding this experience!</p>
 
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@@ -910,33 +1088,29 @@ <h2 id="acknowledgement">Acknowledgement</h2>
 
 
 
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-        <ul class="network-icon" aria-hidden="true">
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+      <p class="card-text">My research interests include many-body physics, probabilistic machine learning, and differentiable programming.</p>
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@@ -1021,8 +1195,36 @@ <h6 class="card-subtitle">Postdoctoral researcher</h6>
   
 </ul>
 
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-      
-      <li><a href="/post/bouncing_ball/">Sensitivity Analysis of Hybrid Differential Equations</a></li>
-      
-      <li><a href="/post/shadowing/">Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></li>
-      
-      <li><a href="/post/hybridde/">Neural Hybrid Differential Equations</a></li>
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+    <article class="article">
 
   
 
@@ -582,6 +788,7 @@ <h1>Search</h1>
 
 
 
+
   
 
   
@@ -625,6 +832,7 @@ <h1>GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</h1>
   
   
   
+  
 
   
   
@@ -637,15 +845,6 @@ <h1>GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</h1>
 
 
 
-
-
-
-
-
-
-
-
-
   
 </div>
 
@@ -654,21 +853,14 @@ <h1>GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</h1>
   <div class="article-container">
 
     <div class="article-style">
-      <p>First and foremost, I would like to thank my mentors 
-<a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, 
-<a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, and  
-<a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a> for their willingness to supervise me in this Google Summer of Code project.
+      <p>First and foremost, I would like to thank my mentors <a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, <a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>, and  <a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a> for their willingness to supervise me in this Google Summer of Code project.
 Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both  <strong>new high weak order solvers</strong> as well as <strong>adjoint sensitivity methods</strong> for stochastic differential equations (SDEs).
-The project is embedded within the 
-<a href="https://sciml.ai" target="_blank" rel="noopener">SciML</a> organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
-Ultimately, the planned contributions will allow researchers to simulate (or even 
-<a href="https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/" target="_blank" rel="noopener">control</a>) stochastic dynamics. Also inverse problems, where 
-<a href="https://diffeqflux.sciml.ai/dev/examples/NN-SDE/" target="_blank" rel="noopener">SDE models are fit to data</a>, fall into the scope.
+The project is embedded within the <a href="https://sciml.ai" target="_blank" rel="noopener">SciML</a> organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
+Ultimately, the planned contributions will allow researchers to simulate (or even <a href="https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/" target="_blank" rel="noopener">control</a>) stochastic dynamics. Also inverse problems, where <a href="https://diffeqflux.sciml.ai/dev/examples/NN-SDE/" target="_blank" rel="noopener">SDE models are fit to data</a>, fall into the scope.
 Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.</p>
 <p>This post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.</p>
 <h2 id="high-weak-order-solvers">High Weak Order Solvers</h2>
-<p>Currently, the 
-<a href="https://github.com/SciML/StochasticDiffEq.jl" target="_blank" rel="noopener">StochasticDiffEq</a> package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
+<p>Currently, the <a href="https://github.com/SciML/StochasticDiffEq.jl" target="_blank" rel="noopener">StochasticDiffEq</a> package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
 In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.</p>
 <p>However in many situations, we are only aiming for computing an estimation for the <strong>expected value of the solution</strong>.
 In such situations, methods for the <strong>weak approximation</strong> are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods.
@@ -676,7 +868,7 @@ <h2 id="high-weak-order-solvers">High Weak Order Solvers</h2>
 As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.</p>
 <h3 id="second-order-runge-kutta-methods-for-ito-sdes">Second order Runge-Kutta methods for Ito SDEs</h3>
 <p>In the beginning of the community bonding period I finished the implementations of the <code>DRI1()</code><sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> and <code>RI1()</code><sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes <code>m</code>. Specifically, only <code>2m-1</code> random variables have to be drawn (in contrast to <code>m(m+1)/2</code> from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of <code>m</code>.</p>
-<p>As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup>:</p>
+<p>As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise<sup id="fnref1:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup>:</p>
 <p>$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} <br>
 $$</p>
@@ -685,47 +877,38 @@ <h3 id="second-order-runge-kutta-methods-for-ito-sdes">Second order Runge-Kutta
 <p>$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$</p>
 <p>which we use to test the weak convergence order of the algorithms in the following.</p>
 <p>To compute the expected value numerically, we sample an ensemble of <code>numtraj = 1e7</code> trajectories for different step sizes <code>dt</code>. The code for a single  <code>dt</code> reads:</p>
-<pre><code class="language-julia">using StochasticDiffEq
-numtraj = 1e7
-u₀ = [1.0,1.0]
-function f!(du,u,p,t)
-  du[1] = -273//512*u[1]
-  du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-end
-function g!(du,u,p,t)
-  du[1,1] = 1//4*u[1]
-  du[1,2] = 1//16*u[1]
-  du[2,1] = (1-2*sqrt(2))/4*u[1]
-  du[2,2] = 1//10*u[1]+1//16*u[2]
-end
-dt = 1//8
-tspan = (0.0,10.0)
-prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-h(z) = z^2
-
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&gt; (h(sol[end][1]),false)
-        )
-sol = solve(ensemble_prob, DRI1();
-        dt=dt,
-        save_start=false,
-        save_everystep=false,
-        weak_timeseries_errors=false,
-        weak_dense_errors=false,
-        trajectories=numtraj)
-
-</code></pre>
-<p>We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:</p>
-<pre><code>LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
-</code></pre>
-<p>Repeating this procedure for some more values of <code>dt</code>, the log-log plot of the error as a function of <code>dt</code> displays nicely the second order convergence (slope $\approx 2.2$).</p>
-
-
-
-
-
-  
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">using</span> <span class="n">StochasticDiffEq</span>
+</span></span><span class="line"><span class="cl"><span class="n">numtraj</span> <span class="o">=</span> <span class="mf">1e7</span>
+</span></span><span class="line"><span class="cl"><span class="n">u₀</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">1.0</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">273</span><span class="o">//</span><span class="mi">512</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="o">//</span><span class="mi">160</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="mi">785</span><span class="o">//</span><span class="mi">512</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">8</span><span class="p">)</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">g!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="o">//</span><span class="mi">4</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="o">//</span><span class="mi">16</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">4</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="o">//</span><span class="mi">10</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="mi">1</span><span class="o">//</span><span class="mi">16</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="n">dt</span> <span class="o">=</span> <span class="mi">1</span><span class="o">//</span><span class="mi">8</span>
+</span></span><span class="line"><span class="cl"><span class="n">tspan</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">10.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob</span> <span class="o">=</span> <span class="n">SDEProblem</span><span class="p">(</span><span class="n">f!</span><span class="p">,</span><span class="n">g!</span><span class="p">,</span><span class="n">u₀</span><span class="p">,</span><span class="n">tspan</span><span class="p">,</span><span class="n">noise_rate_prototype</span><span class="o">=</span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">h</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">=</span> <span class="n">z</span><span class="o">^</span><span class="mi">2</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">ensemble_prob</span> <span class="o">=</span> <span class="n">EnsembleProblem</span><span class="p">(</span><span class="n">prob</span><span class="p">;</span>
+</span></span><span class="line"><span class="cl">        <span class="n">output_func</span> <span class="o">=</span> <span class="p">(</span><span class="n">sol</span><span class="p">,</span><span class="n">i</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="p">(</span><span class="n">h</span><span class="p">(</span><span class="n">sol</span><span class="p">[</span><span class="k">end</span><span class="p">][</span><span class="mi">1</span><span class="p">]),</span><span class="nb">false</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">        <span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">ensemble_prob</span><span class="p">,</span> <span class="n">DRI1</span><span class="p">();</span>
+</span></span><span class="line"><span class="cl">        <span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">        <span class="n">save_start</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">        <span class="n">save_everystep</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">        <span class="n">weak_timeseries_errors</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">        <span class="n">weak_dense_errors</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">        <span class="n">trajectories</span><span class="o">=</span><span class="n">numtraj</span><span class="p">)</span>
+</span></span></code></pre></div><p>We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-fallback" data-lang="fallback"><span class="line"><span class="cl">LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
+</span></span></code></pre></div><p>Repeating this procedure for some more values of <code>dt</code>, the log-log plot of the error as a function of <code>dt</code> displays nicely the second order convergence (slope $\approx 2.2$).</p>
 
 
 
@@ -737,29 +920,26 @@ <h3 id="second-order-runge-kutta-methods-for-ito-sdes">Second order Runge-Kutta
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/DRI1.png" >
 
 
-  <img src="/img/DRI1.png" alt=""  >
-</a>
 
 
 
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="DRI1.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>In the next couple of weeks, my focus will be on</p>
 <ul>
 <li>adding other high weak order solvers,</li>
 <li>implementing adaptive time stepping.</li>
 </ul>
-<p>More of our near-term goals are collected in this 
-<a href="https://github.com/SciML/StochasticDiffEq.jl/issues/182" target="_blank" rel="noopener">issue</a>.</p>
+<p>More of our near-term goals are collected in this <a href="https://github.com/SciML/StochasticDiffEq.jl/issues/182" target="_blank" rel="noopener">issue</a>.</p>
 <h2 id="adjoint-sensitivity-methods-for-sdes">Adjoint Sensitivity Methods for SDEs</h2>
-<p>In 
-<a href="https://mitmath.github.io/18337/lecture10/estimation_identification" target="_blank" rel="noopener">parameter estimation/inverse problems</a>, one is interested to know the optimal choice of parameters <code>p</code> such that a model <code>f(p)</code>, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error</p>
+<p>In <a href="https://mitmath.github.io/18337/lecture10/estimation_identification" target="_blank" rel="noopener">parameter estimation/inverse problems</a>, one is interested to know the optimal choice of parameters <code>p</code> such that a model <code>f(p)</code>, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error</p>
 <p>$$
 L = |f(p)-y|^2.
 $$</p>
@@ -768,47 +948,38 @@ <h2 id="adjoint-sensitivity-methods-for-sdes">Adjoint Sensitivity Methods for SD
 While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.</p>
 <p>Alternatively to the &ldquo;direct&rdquo; AD approaches, the <strong>adjoint sensitivity method</strong> can be used<sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>.
 Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).</p>
-<p>It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an <strong>augmented state backwards in time</strong> launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. <sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/" target="_blank" rel="noopener">DiffEqSensitivity</a> package for ODE adjoints earlier.</p>
+<p>It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an <strong>augmented state backwards in time</strong> launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. <sup id="fnref1:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the <a href="https://github.com/SciML/DiffEqSensitivity.jl/" target="_blank" rel="noopener">DiffEqSensitivity</a> package for ODE adjoints earlier.</p>
 <h3 id="reverse-sde-time-evolution">Reverse SDE time evolution</h3>
 <p>The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms.
 As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed.
 In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values.
-After some fixes for the 
-<a href="https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1" target="_blank" rel="noopener">available noise processes</a>, we are now able to reverse a stochastic time evolution either by using <code>NoiseGrid</code> which linearly interpolates between values of the noise on a given grid or by using a very general <code>NoiseWrapper</code> which interpolates in a distributionally-exact manner based on Brownian bridges.</p>
+After some fixes for the <a href="https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1" target="_blank" rel="noopener">available noise processes</a>, we are now able to reverse a stochastic time evolution either by using <code>NoiseGrid</code> which linearly interpolates between values of the noise on a given grid or by using a very general <code>NoiseWrapper</code> which interpolates in a distributionally-exact manner based on Brownian bridges.</p>
 <p>As an example, the code below computes first the forward evolution of an SDE</p>
 <p>$$ dX  =  \alpha X dt + \beta X dW$$</p>
 <p>with $\alpha=1.01$, $\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.</p>
-<pre><code class="language-julia">  using StochasticDiffEq, DiffEqNoiseProcess
-
-  α=1.01
-  β=0.87
-
-  dt = 1e-3
-  tspan = (0.0,1.0)
-  u₀=1/2
-
-  tarray =  collect(tspan[1]:dt:tspan[2])
-
-  f!(du,u,p,t) = du .= α*u
-  g!(du,u,p,t) = du .= β*u
-
-
-  prob = SDEProblem(f!,g!,[u₀],tspan)
-  sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false)
-
-  _sol = deepcopy(sol) # to make sure the plot is correct
-  W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W))
-  prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1)
-  sol1 = solve(prob1,EulerHeun(),dt=dt)
-</code></pre>
-
-
-
-
-
-  
-
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl">  <span class="k">using</span> <span class="n">StochasticDiffEq</span><span class="p">,</span> <span class="n">DiffEqNoiseProcess</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">α</span><span class="o">=</span><span class="mf">1.01</span>
+</span></span><span class="line"><span class="cl">  <span class="n">β</span><span class="o">=</span><span class="mf">0.87</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">dt</span> <span class="o">=</span> <span class="mf">1e-3</span>
+</span></span><span class="line"><span class="cl">  <span class="n">tspan</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">1.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">u₀</span><span class="o">=</span><span class="mi">1</span><span class="o">/</span><span class="mi">2</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">tarray</span> <span class="o">=</span>  <span class="n">collect</span><span class="p">(</span><span class="n">tspan</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">:</span><span class="n">dt</span><span class="o">:</span><span class="n">tspan</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">f!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span> <span class="o">=</span> <span class="n">du</span> <span class="o">.=</span> <span class="n">α</span><span class="o">*</span><span class="n">u</span>
+</span></span><span class="line"><span class="cl">  <span class="n">g!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span> <span class="o">=</span> <span class="n">du</span> <span class="o">.=</span> <span class="n">β</span><span class="o">*</span><span class="n">u</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">prob</span> <span class="o">=</span> <span class="n">SDEProblem</span><span class="p">(</span><span class="n">f!</span><span class="p">,</span><span class="n">g!</span><span class="p">,[</span><span class="n">u₀</span><span class="p">],</span><span class="n">tspan</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">sol</span> <span class="o">=</span><span class="n">solve</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">EulerHeun</span><span class="p">(),</span><span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">,</span><span class="n">save_noise</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span> <span class="n">adaptive</span><span class="o">=</span><span class="nb">false</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">_sol</span> <span class="o">=</span> <span class="n">deepcopy</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span> <span class="c"># to make sure the plot is correct</span>
+</span></span><span class="line"><span class="cl">  <span class="n">W1</span> <span class="o">=</span> <span class="n">NoiseGrid</span><span class="p">(</span><span class="n">reverse!</span><span class="p">(</span><span class="n">_sol</span><span class="o">.</span><span class="n">t</span><span class="p">),</span><span class="n">reverse!</span><span class="p">(</span><span class="n">_sol</span><span class="o">.</span><span class="n">W</span><span class="o">.</span><span class="n">W</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">  <span class="n">prob1</span> <span class="o">=</span> <span class="n">SDEProblem</span><span class="p">(</span><span class="n">f!</span><span class="p">,</span><span class="n">g!</span><span class="p">,</span><span class="n">sol</span><span class="p">[</span><span class="k">end</span><span class="p">],</span><span class="n">reverse</span><span class="p">(</span><span class="n">tspan</span><span class="p">),</span><span class="n">noise</span><span class="o">=</span><span class="n">W1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">sol1</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob1</span><span class="p">,</span><span class="n">EulerHeun</span><span class="p">(),</span><span class="n">dt</span><span class="o">=</span><span class="n">dt</span><span class="p">)</span>
+</span></span></code></pre></div>
 
 
 
@@ -819,18 +990,17 @@ <h3 id="reverse-sde-time-evolution">Reverse SDE time evolution</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/animation.gif" >
 
 
-  <img src="/img/animation.gif" alt=""  >
-</a>
 
 
 
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="animation.gif" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <h3 id="gradients-of-diagonal-sdes">Gradients of diagonal SDEs</h3>
 <p>I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.</p>
@@ -844,82 +1014,80 @@ <h3 id="gradients-of-diagonal-sdes">Gradients of diagonal SDEs</h3>
 <p>acting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read</p>
 <p>$$
 \begin{aligned}
-\frac{d L}{d \alpha} &amp;= 2 \sum_i t_i |X(t_i)|^2 \\<br>
-\frac{d L}{d \beta}  &amp;= 2 \sum_i W(t_i) |X(t_i)|^2
-\end{aligned}
-$$</p>
-<p>for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method</p>
-<pre><code class="language-julia">using Test, LinearAlgebra
-using DiffEqSensitivity, StochasticDiffEq
-using Random
-
-seed = 100
-Random.seed!(seed)
-
-u₀ = [0.5]
-tstart = 0.0
-tend = 0.1
-dt = 0.005
-trange = (tstart, tend)
-t = tstart:dt:tend
-tarray = collect(t)
-
-function g(u,p,t)
-  sum(u.^2.0)
-end
-
-function dg!(out,u,p,t,i)
-  (out.=-2.0*u)
-end
-
-p2 = [1.01,0.87]
-
-f(u,p,t) = p[1]*u
-σ(u,p,t) = p[2]*u
-
-
-Random.seed!(seed)
-prob = SDEProblem(f,σ,u₀,trange,p2)
-sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true)
-res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint())
-
-
-noise = vec((@. sol.W(tarray)))
-Wextracted = [W[1][1] for W in noise]
-resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp = [resp1, resp2]
-
-@test isapprox(res_p', resp, rtol = 1e-6)
-# True
-</code></pre>
-<p>With respect to the adjoint sensitivity methods, we are planning to</p>
+\frac{d L}{d \alpha} &amp;= 2 \sum_i t_i |X(t_i)|^2 \\
+\frac{d L}{d \beta}  &amp;= 2 \sum_i W(t_i) |X(t_i)|^2
+\end{aligned}
+$$</p>
+<p>for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Test</span><span class="p">,</span> <span class="n">LinearAlgebra</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">DiffEqSensitivity</span><span class="p">,</span> <span class="n">StochasticDiffEq</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Random</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">seed</span> <span class="o">=</span> <span class="mi">100</span>
+</span></span><span class="line"><span class="cl"><span class="n">Random</span><span class="o">.</span><span class="n">seed!</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">u₀</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">tstart</span> <span class="o">=</span> <span class="mf">0.0</span>
+</span></span><span class="line"><span class="cl"><span class="n">tend</span> <span class="o">=</span> <span class="mf">0.1</span>
+</span></span><span class="line"><span class="cl"><span class="n">dt</span> <span class="o">=</span> <span class="mf">0.005</span>
+</span></span><span class="line"><span class="cl"><span class="n">trange</span> <span class="o">=</span> <span class="p">(</span><span class="n">tstart</span><span class="p">,</span> <span class="n">tend</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">t</span> <span class="o">=</span> <span class="n">tstart</span><span class="o">:</span><span class="n">dt</span><span class="ss">:tend</span>
+</span></span><span class="line"><span class="cl"><span class="n">tarray</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">g</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">sum</span><span class="p">(</span><span class="n">u</span><span class="o">.^</span><span class="mf">2.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">dg!</span><span class="p">(</span><span class="n">out</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">i</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="p">(</span><span class="n">out</span><span class="o">.=-</span><span class="mf">2.0</span><span class="o">*</span><span class="n">u</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">p2</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.01</span><span class="p">,</span><span class="mf">0.87</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">f</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">u</span>
+</span></span><span class="line"><span class="cl"><span class="n">σ</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">u</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">Random</span><span class="o">.</span><span class="n">seed!</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob</span> <span class="o">=</span> <span class="n">SDEProblem</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">σ</span><span class="p">,</span><span class="n">u₀</span><span class="p">,</span><span class="n">trange</span><span class="p">,</span><span class="n">p2</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">RKMil</span><span class="p">(</span><span class="n">interpretation</span><span class="o">=</span><span class="ss">:Stratonovich</span><span class="p">),</span><span class="n">dt</span><span class="o">=</span><span class="n">tend</span><span class="o">/</span><span class="mf">1e7</span><span class="p">,</span><span class="n">adaptive</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span><span class="n">save_noise</span><span class="o">=</span><span class="nb">true</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">res_u0</span><span class="p">,</span> <span class="n">res_p</span> <span class="o">=</span> <span class="n">adjoint_sensitivities</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span><span class="n">EulerHeun</span><span class="p">(),</span><span class="n">dg!</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">dt</span><span class="o">=</span><span class="n">tend</span><span class="o">/</span><span class="mf">1e7</span><span class="p">,</span><span class="n">sensealg</span><span class="o">=</span><span class="n">BacksolveAdjoint</span><span class="p">())</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">noise</span> <span class="o">=</span> <span class="n">vec</span><span class="p">((</span><span class="nd">@.</span> <span class="n">sol</span><span class="o">.</span><span class="n">W</span><span class="p">(</span><span class="n">tarray</span><span class="p">)))</span>
+</span></span><span class="line"><span class="cl"><span class="n">Wextracted</span> <span class="o">=</span> <span class="p">[</span><span class="n">W</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">W</span> <span class="k">in</span> <span class="n">noise</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">resp1</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">sum</span><span class="p">(</span><span class="nd">@.</span> <span class="n">tarray</span><span class="o">*</span><span class="n">u₀</span><span class="o">^</span><span class="mi">2</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">p2</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">*</span><span class="n">tarray</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p2</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">Wextracted</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">resp2</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">sum</span><span class="p">(</span><span class="nd">@.</span> <span class="n">Wextracted</span><span class="o">*</span><span class="n">u₀</span><span class="o">^</span><span class="mi">2</span><span class="o">*</span><span class="n">exp</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">p2</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span><span class="o">*</span><span class="n">tarray</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">p2</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">Wextracted</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">resp</span> <span class="o">=</span> <span class="p">[</span><span class="n">resp1</span><span class="p">,</span> <span class="n">resp2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="nd">@test</span> <span class="n">isapprox</span><span class="p">(</span><span class="n">res_p</span><span class="o">&#39;</span><span class="p">,</span> <span class="n">resp</span><span class="p">,</span> <span class="n">rtol</span> <span class="o">=</span> <span class="mf">1e-6</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="c"># True</span>
+</span></span></code></pre></div><p>With respect to the adjoint sensitivity methods, we are planning to</p>
 <ul>
 <li>finish the current backsolve adjoint version,</li>
 <li>allow for computing the gradients of non-commuting SDEs,</li>
 <li>implement also an interpolation adjoint version,</li>
 <li>benchmark it with respect to AD approaches</li>
 </ul>
-<p>in the upcoming weeks. For more information, the interested reader might take a look at the open 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/issues" target="_blank" rel="noopener">issues</a> in the DiffEqSensitivity package.</p>
+<p>in the upcoming weeks. For more information, the interested reader might take a look at the open <a href="https://github.com/SciML/DiffEqSensitivity.jl/issues" target="_blank" rel="noopener">issues</a> in the DiffEqSensitivity package.</p>
 <p>If you have any questions or comments, please don’t hesitate to contact me!</p>
-<section class="footnotes" role="doc-endnotes">
+<div class="footnotes" role="doc-endnotes">
 <hr>
 <ol>
-<li id="fn:1" role="doc-endnote">
-<p>Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics <strong>59</strong>, 582–594 (2009). <a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:1">
+<p>Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics <strong>59</strong>, 582–594 (2009).&#160;<a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:2" role="doc-endnote">
-<p>Andreas Rößler, Journal on Numerical Analysis <strong>47</strong>, 1713–1738 (2009). <a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:2">
+<p>Andreas Rößler, Journal on Numerical Analysis <strong>47</strong>, 1713–1738 (2009).&#160;<a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:3" role="doc-endnote">
-<p>Steven G. Johnson, &ldquo;Notes on Adjoint Methods for 18.335.&rdquo; Introduction to Numerical Methods (2012). <a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:3">
+<p>Steven G. Johnson, &ldquo;Notes on Adjoint Methods for 18.335.&rdquo; Introduction to Numerical Methods (2012).&#160;<a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:4" role="doc-endnote">
-<p>Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). <a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:4">
+<p>Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020).&#160;<a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
 </ol>
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index 4042be8..9b02bf6 100644
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+++ b/post/high-weak/index.html
@@ -1,93 +1,107 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
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-  <meta name="description" content="This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
-After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.">
 
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@@ -656,11 +855,8 @@ <h1>High weak order SDE solvers</h1>
   <div class="article-container">
 
     <div class="article-style">
-      <p>This post summarizes our new 
-<a href="https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1" target="_blank" rel="noopener">high weak order methods</a>
-for the 
-<a href="https://sciml.ai" target="_blank" rel="noopener">SciML</a> ecosystem, as implemented within the 
-<a href="https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112" target="_blank" rel="noopener">Google Summer of Code 2020</a> project.</p>
+      <p>This post summarizes our new <a href="https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1" target="_blank" rel="noopener">high weak order methods</a>
+for the <a href="https://sciml.ai" target="_blank" rel="noopener">SciML</a> ecosystem, as implemented within the <a href="https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112" target="_blank" rel="noopener">Google Summer of Code 2020</a> project.</p>
 <p>After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
 Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics.
 Finally, we discuss how to run simulations on GPU hardware.</p>
@@ -691,16 +887,14 @@ <h2 id="strong-convergence">Strong convergence</h2>
 \rm{E}(|X(T) -Y(T)|) \leq C \cdot h^p,
 $$
 for each $h \in [0, \delta]$.</p>
-<p>The 
-<a href="https://github.com/SciML/StochasticDiffEq.jl" target="_blank" rel="noopener">StochasticDiffEq</a> package contains various state-of-the-art solvers
+<p>The <a href="https://github.com/SciML/StochasticDiffEq.jl" target="_blank" rel="noopener">StochasticDiffEq</a> package contains various state-of-the-art solvers
 for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms.
 For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI)
 can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where <em>b</em> has only entries on its diagonal or $m=1$.
 The main difficulty for the construction of strong methods with an order &gt; 1/2 arises from the need of an accurate estimation of
 multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of <em>dW</em> in the case
 of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed
-in the case of non-commutative noise processes.<sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> Currently, we are also 
-<a href="https:://github.com/SciML/StochasticDiffEq.jl/pull/347" target="_blank" rel="noopener">implementing those iterated integrals in the StochasticDiffEq library</a>.</p>
+in the case of non-commutative noise processes.<sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> Currently, we are also <a href="https:://github.com/SciML/StochasticDiffEq.jl/pull/347" target="_blank" rel="noopener">implementing those iterated integrals in the StochasticDiffEq library</a>.</p>
 <h2 id="weak-convergence">Weak convergence</h2>
 <p>Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the <strong>expected value of the solution</strong> in many situations. In these cases, methods for the <strong>weak approximation</strong> are sufficient and &ndash; due to the less restrictive formulation of the objective &ndash; those solvers are computationally cheaper than their strong counterparts.
 For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics.
@@ -715,19 +909,17 @@ <h2 id="weak-convergence">Weak convergence</h2>
 <p>for any polynomial $g$ for each $h \in [0, \delta]$.</p>
 <p>We demonstrate below that <strong>high weak order solvers</strong> are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.</p>
 <h2 id="new-high-weak-order-methods">New high weak order methods</h2>
-<p>A list of all new weak solvers is available in the 
-<a href="https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1" target="_blank" rel="noopener">SciML documentation</a>.
+<p>A list of all new weak solvers is available in the <a href="https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1" target="_blank" rel="noopener">SciML documentation</a>.
 Note that we also implemented methods designed for the Stratonovich sense.
 For the subsequent examples regarding Ito SDEs, we use only a subset of the plethora of second-order weak solvers.
-We employ the <code>DRI1()</code><sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>, <code>RD1WM()</code><sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>, and <code>RD2WM()</code><sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> methods due to Debrabant &amp; Rößler and Platen&rsquo;s <code>PL1WM()</code><sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> method.
-We compare those methods to the strong Euler-Maruyama <code>EM()</code><sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> and the simplified Euler-Maruyama <code>SimplifiedEM()</code><sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> schemes.
+We employ the <code>DRI1()</code><sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>, <code>RD1WM()</code><sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup>, and <code>RD2WM()</code><sup id="fnref1:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> methods due to Debrabant &amp; Rößler and Platen&rsquo;s <code>PL1WM()</code><sup id="fnref1:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> method.
+We compare those methods to the strong Euler-Maruyama <code>EM()</code><sup id="fnref2:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> and the simplified Euler-Maruyama <code>SimplifiedEM()</code><sup id="fnref3:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> schemes.
 The latter is the simplest weak solver, where the Gaussian increments of the strong Euler-Maruyama scheme are replaced by
 two-point distributed random variables with similar moment properties.</p>
 <p>Rößler&rsquo;s SRK schemes are particularly designed to scale well with the number of Wiener processes <code>m</code>, since only <code>2m-1</code> random variables have to be drawn and since the number of function evaluations for the drift and the diffusion terms is independent of <code>m</code>.
 <code>PL1WM()</code> in contrast needs to simulate <code>m(m+1)/2</code> random variables but a smaller number of order conditions needs to be fulfilled.</p>
 <h3 id="convergence-tests">Convergence tests</h3>
-<p>As in the 
-<a href="https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" target="_blank" rel="noopener">first blog post</a>, let us consider the multi-dimensional SDE with non-commuting noise<sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>:</p>
+<p>As in the <a href="https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" target="_blank" rel="noopener">first blog post</a>, let us consider the multi-dimensional SDE with non-commuting noise<sup id="fnref1:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>:</p>
 <p>$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} <br>
 $$</p>
@@ -738,67 +930,61 @@ <h3 id="convergence-tests">Convergence tests</h3>
 <p>$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$</p>
 <p>for the function $f(x)=(x_1)^2$, which we use to test the weak convergence order of the algorithms in the following.</p>
 <p>To compute the expected value numerically, we sample an ensemble of <code>numtraj = 1e6</code> trajectories for different step sizes <code>dt</code>.</p>
-<pre><code class="language-julia">using StochasticDiffEq
-using Test
-using Random
-using Plots
-using DiffEqDevTools
-
-function prob_func(prob, i, repeat)
-    remake(prob,seed=seeds[i])
-end
-
-u₀ = [1.0,1.0]
-function f1!(du,u,p,t)
-  @inbounds begin
-    du[1] = -273//512*u[1]
-    du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-  end
-  return nothing
-end
-function g1!(du,u,p,t)
-  @inbounds begin
-    du[1,1] = 1//4*u[1]
-    du[1,2] = 1//16*u[1]
-    du[2,1] = (1-2*sqrt(2))/4*u[1]
-    du[2,2] = 1//10*u[1]+1//16*u[2]
-  end
-  return nothing
-end
-dts = 1 .//2 .^(3: -1:0)
-tspan = (0.0,3.0)
-
-h2(z) = z^2 # but apply it only to u[1]
-
-prob = SDEProblem(f1!,g1!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-numtraj = Int(1e6)
-seed = 100
-Random.seed!(seed)
-seeds = rand(UInt, numtraj)
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&gt; (h2(sol[end][1]),false),
-        prob_func = prob_func
-        )
-
-sim = test_convergence(dts,ensemble_prob,DRI1(),
-  save_everystep=false,trajectories=numtraj,save_start=false,adaptive=false,
-  weak_timeseries_errors=false,weak_dense_errors=false,
-  expected_value=exp(-3.0)
-  )
-</code></pre>
-<p>The object <code>sim</code> defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the <code>DRI1()</code> scheme.
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">using</span> <span class="n">StochasticDiffEq</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Test</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Random</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Plots</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">DiffEqDevTools</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">prob_func</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">repeat</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">remake</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">seed</span><span class="o">=</span><span class="n">seeds</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">u₀</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">1.0</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f1!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="nd">@inbounds</span> <span class="k">begin</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">273</span><span class="o">//</span><span class="mi">512</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span><span class="o">//</span><span class="mi">160</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="p">(</span><span class="o">-</span><span class="mi">785</span><span class="o">//</span><span class="mi">512</span><span class="o">+</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">8</span><span class="p">)</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="k">end</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">g1!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="nd">@inbounds</span> <span class="k">begin</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="o">//</span><span class="mi">4</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="o">//</span><span class="mi">16</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span><span class="o">/</span><span class="mi">4</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="o">//</span><span class="mi">10</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="mi">1</span><span class="o">//</span><span class="mi">16</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="k">end</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="n">dts</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">.//</span><span class="mi">2</span> <span class="o">.^</span><span class="p">(</span><span class="mi">3</span><span class="o">:</span> <span class="o">-</span><span class="mi">1</span><span class="o">:</span><span class="mi">0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">tspan</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">3.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">h2</span><span class="p">(</span><span class="n">z</span><span class="p">)</span> <span class="o">=</span> <span class="n">z</span><span class="o">^</span><span class="mi">2</span> <span class="c"># but apply it only to u[1]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">prob</span> <span class="o">=</span> <span class="n">SDEProblem</span><span class="p">(</span><span class="n">f1!</span><span class="p">,</span><span class="n">g1!</span><span class="p">,</span><span class="n">u₀</span><span class="p">,</span><span class="n">tspan</span><span class="p">,</span><span class="n">noise_rate_prototype</span><span class="o">=</span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">numtraj</span> <span class="o">=</span> <span class="kt">Int</span><span class="p">(</span><span class="mf">1e6</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">seed</span> <span class="o">=</span> <span class="mi">100</span>
+</span></span><span class="line"><span class="cl"><span class="n">Random</span><span class="o">.</span><span class="n">seed!</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">seeds</span> <span class="o">=</span> <span class="n">rand</span><span class="p">(</span><span class="kt">UInt</span><span class="p">,</span> <span class="n">numtraj</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">ensemble_prob</span> <span class="o">=</span> <span class="n">EnsembleProblem</span><span class="p">(</span><span class="n">prob</span><span class="p">;</span>
+</span></span><span class="line"><span class="cl">        <span class="n">output_func</span> <span class="o">=</span> <span class="p">(</span><span class="n">sol</span><span class="p">,</span><span class="n">i</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="p">(</span><span class="n">h2</span><span class="p">(</span><span class="n">sol</span><span class="p">[</span><span class="k">end</span><span class="p">][</span><span class="mi">1</span><span class="p">]),</span><span class="nb">false</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">        <span class="n">prob_func</span> <span class="o">=</span> <span class="n">prob_func</span>
+</span></span><span class="line"><span class="cl">        <span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">sim</span> <span class="o">=</span> <span class="n">test_convergence</span><span class="p">(</span><span class="n">dts</span><span class="p">,</span><span class="n">ensemble_prob</span><span class="p">,</span><span class="n">DRI1</span><span class="p">(),</span>
+</span></span><span class="line"><span class="cl">  <span class="n">save_everystep</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span><span class="n">trajectories</span><span class="o">=</span><span class="n">numtraj</span><span class="p">,</span><span class="n">save_start</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span><span class="n">adaptive</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">weak_timeseries_errors</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span><span class="n">weak_dense_errors</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">expected_value</span><span class="o">=</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mf">3.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="p">)</span>
+</span></span></code></pre></div><p>The object <code>sim</code> defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the <code>DRI1()</code> scheme.
 Repeating this call to the <code>test_convergence()</code> function for the other aforementioned solvers, we obtain the convergence plot:</p>
 
 
 
 
 
-  
-
-
-
-
 
 
 
@@ -806,18 +992,16 @@ <h3 id="convergence-tests">Convergence tests</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/weak_conv.png" >
 
 
-  <img src="/img/weak_conv.png" alt=""  >
-</a>
 
 
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="weak_conv.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>Note that the <code>SimplifiedEM</code> and the <code>EM</code> scheme fall on top of each other.
 <code>DRI1()</code> achieves the smallest errors for a fixed <code>dt</code> in this study.</p>
@@ -825,35 +1009,28 @@ <h3 id="work-precision-diagrams">Work-Precision Diagrams</h3>
 <p>Ultimately, we are not only interested in the general convergence slope of an algorithm but also in its speed.
 We&rsquo;d like to select the fastest method depending on the permitted tolerance.
 This is commonly studied using work-precision diagram.
-Thanks to 
-<a href="https://github.com/SciML/DiffEqDevTools.jl/pull/75" target="_blank" rel="noopener">new routines</a>, a user can generate a work-precision diagram by the following code</p>
-<pre><code class="language-julia">reltols = 1.0 ./ 4.0 .^ (1:4)
-abstols = reltols#[0.0 for i in eachindex(reltols)]
-setups = [
-          Dict(:alg=&gt;DRI1(),:dts=&gt;dts,:adaptive=&gt;false),
-          Dict(:alg=&gt;PL1WM(),:dts=&gt;dts,:adaptive=&gt;false),
-          Dict(:alg=&gt;EM(),:dts=&gt;dts,:adaptive=&gt;false),
-          Dict(:alg=&gt;SimplifiedEM(),:dts=&gt;dts,:adaptive=&gt;false),
-          Dict(:alg=&gt;RDI2WM(),:dts=&gt;dts,:adaptive=&gt;false),
-          Dict(:alg=&gt;RDI1WM(),:dts=&gt;dts,:adaptive=&gt;false)
-          ]
-test_dt = 1//10000
-appxsol_setup = Dict(:alg=&gt;EM(), :dt=&gt;test_dt)
-wp = @time WorkPrecisionSet(ensemble_prob,
-                            abstols,reltols,setups,test_dt;
-                            maxiters = 1e7,verbose=false,
-                            save_everystep=false,save_start=false,
-                            appxsol_setup = appxsol_setup,
-                            expected_value=exp(-3.0),
-                            trajectories=numtraj, error_estimate=:weak_final)
-plt = plot(wp;legend=:bottomleft)
-</code></pre>
-
-
-
-
-
-  
+Thanks to <a href="https://github.com/SciML/DiffEqDevTools.jl/pull/75" target="_blank" rel="noopener">new routines</a>, a user can generate a work-precision diagram by the following code</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="n">reltols</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">./</span> <span class="mf">4.0</span> <span class="o">.^</span> <span class="p">(</span><span class="mi">1</span><span class="o">:</span><span class="mi">4</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">abstols</span> <span class="o">=</span> <span class="n">reltols</span><span class="c">#[0.0 for i in eachindex(reltols)]</span>
+</span></span><span class="line"><span class="cl"><span class="n">setups</span> <span class="o">=</span> <span class="p">[</span>
+</span></span><span class="line"><span class="cl">          <span class="kt">Dict</span><span class="p">(</span><span class="ss">:alg</span><span class="o">=&gt;</span><span class="n">DRI1</span><span class="p">(),</span><span class="ss">:dts</span><span class="o">=&gt;</span><span class="n">dts</span><span class="p">,</span><span class="ss">:adaptive</span><span class="o">=&gt;</span><span class="nb">false</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">          <span class="kt">Dict</span><span class="p">(</span><span class="ss">:alg</span><span class="o">=&gt;</span><span class="n">PL1WM</span><span class="p">(),</span><span class="ss">:dts</span><span class="o">=&gt;</span><span class="n">dts</span><span class="p">,</span><span class="ss">:adaptive</span><span class="o">=&gt;</span><span class="nb">false</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">          <span class="kt">Dict</span><span class="p">(</span><span class="ss">:alg</span><span class="o">=&gt;</span><span class="n">EM</span><span class="p">(),</span><span class="ss">:dts</span><span class="o">=&gt;</span><span class="n">dts</span><span class="p">,</span><span class="ss">:adaptive</span><span class="o">=&gt;</span><span class="nb">false</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">          <span class="kt">Dict</span><span class="p">(</span><span class="ss">:alg</span><span class="o">=&gt;</span><span class="n">SimplifiedEM</span><span class="p">(),</span><span class="ss">:dts</span><span class="o">=&gt;</span><span class="n">dts</span><span class="p">,</span><span class="ss">:adaptive</span><span class="o">=&gt;</span><span class="nb">false</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">          <span class="kt">Dict</span><span class="p">(</span><span class="ss">:alg</span><span class="o">=&gt;</span><span class="n">RDI2WM</span><span class="p">(),</span><span class="ss">:dts</span><span class="o">=&gt;</span><span class="n">dts</span><span class="p">,</span><span class="ss">:adaptive</span><span class="o">=&gt;</span><span class="nb">false</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">          <span class="kt">Dict</span><span class="p">(</span><span class="ss">:alg</span><span class="o">=&gt;</span><span class="n">RDI1WM</span><span class="p">(),</span><span class="ss">:dts</span><span class="o">=&gt;</span><span class="n">dts</span><span class="p">,</span><span class="ss">:adaptive</span><span class="o">=&gt;</span><span class="nb">false</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">          <span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">test_dt</span> <span class="o">=</span> <span class="mi">1</span><span class="o">//</span><span class="mi">10000</span>
+</span></span><span class="line"><span class="cl"><span class="n">appxsol_setup</span> <span class="o">=</span> <span class="kt">Dict</span><span class="p">(</span><span class="ss">:alg</span><span class="o">=&gt;</span><span class="n">EM</span><span class="p">(),</span> <span class="ss">:dt</span><span class="o">=&gt;</span><span class="n">test_dt</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">wp</span> <span class="o">=</span> <span class="nd">@time</span> <span class="n">WorkPrecisionSet</span><span class="p">(</span><span class="n">ensemble_prob</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">                            <span class="n">abstols</span><span class="p">,</span><span class="n">reltols</span><span class="p">,</span><span class="n">setups</span><span class="p">,</span><span class="n">test_dt</span><span class="p">;</span>
+</span></span><span class="line"><span class="cl">                            <span class="n">maxiters</span> <span class="o">=</span> <span class="mf">1e7</span><span class="p">,</span><span class="n">verbose</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">                            <span class="n">save_everystep</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span><span class="n">save_start</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">                            <span class="n">appxsol_setup</span> <span class="o">=</span> <span class="n">appxsol_setup</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">                            <span class="n">expected_value</span><span class="o">=</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="mf">3.0</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">                            <span class="n">trajectories</span><span class="o">=</span><span class="n">numtraj</span><span class="p">,</span> <span class="n">error_estimate</span><span class="o">=</span><span class="ss">:weak_final</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">plt</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">wp</span><span class="p">;</span><span class="n">legend</span><span class="o">=</span><span class="ss">:bottomleft</span><span class="p">)</span>
+</span></span></code></pre></div>
 
 
 
@@ -865,91 +1042,77 @@ <h3 id="work-precision-diagrams">Work-Precision Diagrams</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/WorkPrecision.png" >
 
 
-  <img src="/img/WorkPrecision.png" alt=""  >
-</a>
 
 
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="WorkPrecision.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>Therefore, <code>DRI1</code> has the best performance in this non-commutative noise case if the error is supposed to stay below 1e-3.
 For larger permitted errors, the <code>SimplifiedEM</code> scheme might be a good choice. However, the first order methods
 are outclassed when high precision is more of a concern.
-We plan to perform more in-depth benchmarks in the near future what will be reported on the 
-<a href="https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes" target="_blank" rel="noopener">SciML news</a>. Stay tuned!</p>
+We plan to perform more in-depth benchmarks in the near future what will be reported on the <a href="https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes" target="_blank" rel="noopener">SciML news</a>. Stay tuned!</p>
 <h3 id="adaptive-step-size-control">Adaptive step-size control</h3>
-<p>Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.</p>
+<p>Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.<sup id="fnref2:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.</p>
 <p>To use adaptive step-size control, it is sufficient to set <code>adaptive=true</code> (default setting). Optionally, one may also pass absolute and relative tolerances.</p>
 <p>The following julia code simulates the stochastic version of the Brusselator equations with intitial condition</p>
 <p>$$ X(t=0)=  \begin{pmatrix} 0.1 \\ 0\end{pmatrix},$$</p>
 <p>on a time span $\mathbb{I}=[0, 100]$ for adaptive (<code>sol</code>) and fixed step sizes (<code>sol_na</code>):</p>
-<pre><code class="language-julia">using StochasticDiffEq, DiffEqNoiseProcess, Random
-using Plots
-using DiffEqGPU
-
-function prob_func(prob, i, repeat)
-  Random.seed!(seeds[i])
-  W = WienerProcess(0.0,0.0,0.0)
-  remake(prob,noise=W)
-end
-
-function brusselator_f!(du,u,p,t)
-  @inbounds begin
-    du[1] = (p[1]-1)*u[1]+p[1]*u[1]^2+(u[1]+1)^2*u[2]
-    du[2] = -p[1]*u[1]-p[1]*u[1]^2-(u[1]+1)^2*u[2]
-  end
-  nothing
-end
-
-function scalar_noise!(du,u,p,t)
-  @inbounds begin
-    du[1] = p[2]*u[1]*(1+u[1])
-    du[2] = -p[2]*u[1]*(1+u[1])
-   end
-   nothing
-end
-
-
-# fix seeds
-seed = 100
-Random.seed!(seed)
-numtraj= 100
-seeds = rand(UInt, numtraj)
-W = WienerProcess(0.0,0.0,0.0)
-
-
-u0 = [-0.1f0,0.0f0]
-tspan = (0.0f0,100.0f0)
-p = [1.9f0,0.1f0]
-
-prob = SDEProblem(brusselator_f!,scalar_noise!,u0,tspan,p,noise=W)
-
-ensembleprob = EnsembleProblem(prob, prob_func = prob_func)
-
-sol = @time solve(ensembleprob,DRI1(),dt=0.1,EnsembleCPUArray(),trajectories=numtraj)
-sol_na = @time solve(ensembleprob,DRI1(),dt=0.8,adaptive=false,EnsembleCPUArray(),trajectories=numtraj)
-
-
-summ = EnsembleSummary(sol,0.0f0:0.5f0:100f0)
-pl = plot(summ,fillalpha=0.5,xlabel = &quot;time t&quot;, yaxis=&quot;X(t)&quot;, label= [&quot;x₁(t)&quot; &quot;x₂(t)&quot;], legend=true)
-
-</code></pre>
-<p>The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.</p>
-
-
-
-
-
-  
-
-
-
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">using</span> <span class="n">StochasticDiffEq</span><span class="p">,</span> <span class="n">DiffEqNoiseProcess</span><span class="p">,</span> <span class="n">Random</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Plots</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">DiffEqGPU</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">prob_func</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">repeat</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">Random</span><span class="o">.</span><span class="n">seed!</span><span class="p">(</span><span class="n">seeds</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">  <span class="n">W</span> <span class="o">=</span> <span class="n">WienerProcess</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">remake</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">noise</span><span class="o">=</span><span class="n">W</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">brusselator_f!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="nd">@inbounds</span> <span class="k">begin</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">^</span><span class="mi">2</span><span class="o">+</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">^</span><span class="mi">2</span><span class="o">-</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="k">end</span>
+</span></span><span class="line"><span class="cl">  <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">scalar_noise!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="nd">@inbounds</span> <span class="k">begin</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">   <span class="k">end</span>
+</span></span><span class="line"><span class="cl">   <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># fix seeds</span>
+</span></span><span class="line"><span class="cl"><span class="n">seed</span> <span class="o">=</span> <span class="mi">100</span>
+</span></span><span class="line"><span class="cl"><span class="n">Random</span><span class="o">.</span><span class="n">seed!</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">numtraj</span><span class="o">=</span> <span class="mi">100</span>
+</span></span><span class="line"><span class="cl"><span class="n">seeds</span> <span class="o">=</span> <span class="n">rand</span><span class="p">(</span><span class="kt">UInt</span><span class="p">,</span> <span class="n">numtraj</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">W</span> <span class="o">=</span> <span class="n">WienerProcess</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">0.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">u0</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.1f0</span><span class="p">,</span><span class="mf">0.0f0</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">tspan</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.0f0</span><span class="p">,</span><span class="mf">100.0f0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.9f0</span><span class="p">,</span><span class="mf">0.1f0</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">prob</span> <span class="o">=</span> <span class="n">SDEProblem</span><span class="p">(</span><span class="n">brusselator_f!</span><span class="p">,</span><span class="n">scalar_noise!</span><span class="p">,</span><span class="n">u0</span><span class="p">,</span><span class="n">tspan</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">noise</span><span class="o">=</span><span class="n">W</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">ensembleprob</span> <span class="o">=</span> <span class="n">EnsembleProblem</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span> <span class="n">prob_func</span> <span class="o">=</span> <span class="n">prob_func</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">sol</span> <span class="o">=</span> <span class="nd">@time</span> <span class="n">solve</span><span class="p">(</span><span class="n">ensembleprob</span><span class="p">,</span><span class="n">DRI1</span><span class="p">(),</span><span class="n">dt</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span><span class="n">EnsembleCPUArray</span><span class="p">(),</span><span class="n">trajectories</span><span class="o">=</span><span class="n">numtraj</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol_na</span> <span class="o">=</span> <span class="nd">@time</span> <span class="n">solve</span><span class="p">(</span><span class="n">ensembleprob</span><span class="p">,</span><span class="n">DRI1</span><span class="p">(),</span><span class="n">dt</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span><span class="n">adaptive</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span><span class="n">EnsembleCPUArray</span><span class="p">(),</span><span class="n">trajectories</span><span class="o">=</span><span class="n">numtraj</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">summ</span> <span class="o">=</span> <span class="n">EnsembleSummary</span><span class="p">(</span><span class="n">sol</span><span class="p">,</span><span class="mf">0.0f0</span><span class="o">:</span><span class="mf">0.5f0</span><span class="o">:</span><span class="mf">100f0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">summ</span><span class="p">,</span><span class="n">fillalpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span><span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;time t&#34;</span><span class="p">,</span> <span class="n">yaxis</span><span class="o">=</span><span class="s">&#34;X(t)&#34;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span> <span class="p">[</span><span class="s">&#34;x₁(t)&#34;</span> <span class="s">&#34;x₂(t)&#34;</span><span class="p">],</span> <span class="n">legend</span><span class="o">=</span><span class="nb">true</span><span class="p">)</span>
+</span></span></code></pre></div><p>The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.</p>
 
 
 
@@ -958,69 +1121,63 @@ <h3 id="adaptive-step-size-control">Adaptive step-size control</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/Brusselator_many_trajectories.png" >
-
-
-  <img src="/img/Brusselator_many_trajectories.png" alt=""  >
-</a>
-
-
-
-</figure>
-
-<p>We can confirm Rößler&rsquo;s observation<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> that the adaptive scheme describes the time evolution of the SDE more accurately,
-as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.</p>
-<pre><code class="language-julia">using DifferentialEquations.EnsembleAnalysis
-meansol = timeseries_steps_mean(sol)
-meansol_na = timeseries_point_mean(sol_na,meansol.t)
-dts = []
-tmp1 = tspan[1]
-for tmp2 in meansol.t
-  global tmp1
-  push!(dts,tmp2-tmp1)
-  tmp1 = tmp2
-end
-#
 
-list_plots = []
-for i in 1:length(meansol.u)
-  l = @layout [a  b]
-  plt1 = plot(meansol[1, 1:i],meansol[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &quot;x₁(t)&quot;,
-         yaxis= &quot;x₂(t)&quot;,
-         label=&quot;adaptive&quot;,
-         lw=2,
-         linecolor=1
-         )
-  plot!(meansol_na[1, 1:i],meansol_na[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &quot;x₁(t)&quot;,
-         yaxis= &quot;x₂(t)&quot;,
-         label=&quot;fixed step size&quot;,
-         lw=2,
-         linecolor=2
-         )
-  pl2 = scatter(dts[1:i], xlabel = &quot;step&quot;, yaxis= &quot;dtᵢ&quot;, xlim = (0, length(meansol.u)),  ylim = (0.0, 2.3), legend=false)
-  plt = plot(plt1, pl2, layout = l)
-  push!(list_plots, plt)
-end
 
-anim = animate(list_plots,lw=2,every=1)
 
-</code></pre>
 
 
 
 
 
-  
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="Brusselator_many_trajectories.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
+<p>We can confirm Rößler&rsquo;s observation<sup id="fnref3:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> that the adaptive scheme describes the time evolution of the SDE more accurately,
+as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">using</span> <span class="n">DifferentialEquations</span><span class="o">.</span><span class="n">EnsembleAnalysis</span>
+</span></span><span class="line"><span class="cl"><span class="n">meansol</span> <span class="o">=</span> <span class="n">timeseries_steps_mean</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">meansol_na</span> <span class="o">=</span> <span class="n">timeseries_point_mean</span><span class="p">(</span><span class="n">sol_na</span><span class="p">,</span><span class="n">meansol</span><span class="o">.</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">dts</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl"><span class="n">tmp1</span> <span class="o">=</span> <span class="n">tspan</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">for</span> <span class="n">tmp2</span> <span class="k">in</span> <span class="n">meansol</span><span class="o">.</span><span class="n">t</span>
+</span></span><span class="line"><span class="cl">  <span class="k">global</span> <span class="n">tmp1</span>
+</span></span><span class="line"><span class="cl">  <span class="n">push!</span><span class="p">(</span><span class="n">dts</span><span class="p">,</span><span class="n">tmp2</span><span class="o">-</span><span class="n">tmp1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">tmp1</span> <span class="o">=</span> <span class="n">tmp2</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="c">#</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">list_plots</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl"><span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="mi">1</span><span class="o">:</span><span class="n">length</span><span class="p">(</span><span class="n">meansol</span><span class="o">.</span><span class="n">u</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">l</span> <span class="o">=</span> <span class="nd">@layout</span> <span class="p">[</span><span class="n">a</span>  <span class="n">b</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">plt1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">meansol</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="o">:</span><span class="n">i</span><span class="p">],</span><span class="n">meansol</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="o">:</span><span class="n">i</span><span class="p">],</span>
+</span></span><span class="line"><span class="cl">         <span class="n">ylim</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mf">0.18</span><span class="p">,</span> <span class="mf">0.18</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">         <span class="n">xlim</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mf">0.13</span><span class="p">,</span> <span class="mf">0.13</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">         <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;x₁(t)&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">         <span class="n">yaxis</span><span class="o">=</span> <span class="s">&#34;x₂(t)&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">         <span class="n">label</span><span class="o">=</span><span class="s">&#34;adaptive&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">         <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">         <span class="n">linecolor</span><span class="o">=</span><span class="mi">1</span>
+</span></span><span class="line"><span class="cl">         <span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">plot!</span><span class="p">(</span><span class="n">meansol_na</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="o">:</span><span class="n">i</span><span class="p">],</span><span class="n">meansol_na</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="o">:</span><span class="n">i</span><span class="p">],</span>
+</span></span><span class="line"><span class="cl">         <span class="n">ylim</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mf">0.18</span><span class="p">,</span> <span class="mf">0.18</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">         <span class="n">xlim</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mf">0.13</span><span class="p">,</span> <span class="mf">0.13</span><span class="p">),</span>
+</span></span><span class="line"><span class="cl">         <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;x₁(t)&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">         <span class="n">yaxis</span><span class="o">=</span> <span class="s">&#34;x₂(t)&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">         <span class="n">label</span><span class="o">=</span><span class="s">&#34;fixed step size&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">         <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">         <span class="n">linecolor</span><span class="o">=</span><span class="mi">2</span>
+</span></span><span class="line"><span class="cl">         <span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">pl2</span> <span class="o">=</span> <span class="n">scatter</span><span class="p">(</span><span class="n">dts</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="n">i</span><span class="p">],</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;step&#34;</span><span class="p">,</span> <span class="n">yaxis</span><span class="o">=</span> <span class="s">&#34;dtᵢ&#34;</span><span class="p">,</span> <span class="n">xlim</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">length</span><span class="p">(</span><span class="n">meansol</span><span class="o">.</span><span class="n">u</span><span class="p">)),</span>  <span class="n">ylim</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">2.3</span><span class="p">),</span> <span class="n">legend</span><span class="o">=</span><span class="nb">false</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">plt</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">plt1</span><span class="p">,</span> <span class="n">pl2</span><span class="p">,</span> <span class="n">layout</span> <span class="o">=</span> <span class="n">l</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">push!</span><span class="p">(</span><span class="n">list_plots</span><span class="p">,</span> <span class="n">plt</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">anim</span> <span class="o">=</span> <span class="n">animate</span><span class="p">(</span><span class="n">list_plots</span><span class="p">,</span><span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span><span class="n">every</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+</span></span></code></pre></div>
 
 
 
@@ -1031,53 +1188,49 @@ <h3 id="adaptive-step-size-control">Adaptive step-size control</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/Brusselator.gif" >
 
 
-  <img src="/img/Brusselator.gif" alt=""  >
-</a>
 
 
 
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="Brusselator.gif" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <h3 id="gpu-usage">GPU usage</h3>
 <p>All necessary tools to accelerate the simulation of (stochastic) differential equations on GPUs within the SciML
-ecosystem are collected in the 
-<a href="https://github.com/SciML/DiffEqGPU.jl" target="_blank" rel="noopener">DiffEqGPU</a> package.</p>
+ecosystem are collected in the <a href="https://github.com/SciML/DiffEqGPU.jl" target="_blank" rel="noopener">DiffEqGPU</a> package.</p>
 <p>Currently, bounds checking and return values are not allowed, i.e., functions must be
 written in the form:</p>
-<pre><code class="language-julia">function f!(du,u,p,t)
-  @inbounds begin
-    du[1] = ..
-  end
-  nothing
-end
-</code></pre>
-<p>Except from those limitations, a user can specifiy <code>ensemblealg=EnsembleGPUArray()</code> to parallelize SDE solves across the GPU, see, e.g., 
-<a href="https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl" target="_blank" rel="noopener">the GPU tests for StochasticDiffEq</a> for some examples.
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="nd">@inbounds</span> <span class="k">begin</span>
+</span></span><span class="line"><span class="cl">    <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="o">..</span>
+</span></span><span class="line"><span class="cl">  <span class="k">end</span>
+</span></span><span class="line"><span class="cl">  <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><p>Except from those limitations, a user can specifiy <code>ensemblealg=EnsembleGPUArray()</code> to parallelize SDE solves across the GPU, see, e.g., <a href="https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl" target="_blank" rel="noopener">the GPU tests for StochasticDiffEq</a> for some examples.
 Note that for some high weak order solvers GPU usage is not recommended as scalar indexing is used.</p>
 <p>If you have any questions or comments, please don’t hesitate to contact me!</p>
-<section class="footnotes" role="doc-endnotes">
+<div class="footnotes" role="doc-endnotes">
 <hr>
 <ol>
-<li id="fn:1" role="doc-endnote">
-<p>Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. <strong>23</strong>, Springer Science &amp; Business Media (2013). <a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:1">
+<p>Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. <strong>23</strong>, Springer Science &amp; Business Media (2013).&#160;<a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref2:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref3:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:2" role="doc-endnote">
-<p>Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications <strong>10</strong> 431-441 (1992). <a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:2">
+<p>Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications <strong>10</strong> 431-441 (1992).&#160;<a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:3" role="doc-endnote">
-<p>Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics <strong>59</strong>, 582–594 (2009). <a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:3">
+<p>Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics <strong>59</strong>, 582–594 (2009).&#160;<a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:4" role="doc-endnote">
-<p>Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation <strong>77</strong>, 408-420 (2008) <a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:4">
+<p>Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation <strong>77</strong>, 408-420 (2008)&#160;<a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref2:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref3:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
 </ol>
-</section>
+</div>
 
     </div>
 
@@ -1091,17 +1244,17 @@ <h3 id="gpu-usage">GPU usage</h3>
   
   <a class="badge badge-light" href="/tag/gsoc-2020/">GSoC 2020</a>
   
-  <a class="badge badge-light" href="/tag/julia/">julia</a>
+  <a class="badge badge-light" href="/tag/julia/">Julia</a>
   
-  <a class="badge badge-light" href="/tag/high-weak-order-solver/">High weak order solver</a>
+  <a class="badge badge-light" href="/tag/high-weak-order-solver/">High Weak Order Solver</a>
   
-  <a class="badge badge-light" href="/tag/srk-methods/">SRK methods</a>
+  <a class="badge badge-light" href="/tag/srk-methods/">SRK Methods</a>
   
 </div>
 
 
 
-<div class="share-box" aria-hidden="true">
+<div class="share-box">
   <ul class="share">
     
       
@@ -1111,8 +1264,12 @@ <h3 id="gpu-usage">GPU usage</h3>
       
       
       
+      
+      
+      
+      
       <li>
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+        <a href="https://twitter.com/intent/tweet?url=https%3A%2F%2Ffrankschae.github.io%2Fpost%2Fhigh-weak%2F&amp;text=High&#43;weak&#43;order&#43;SDE&#43;solvers" target="_blank" rel="noopener" class="share-btn-twitter" aria-label="twitter">
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@@ -1124,8 +1281,12 @@ <h3 id="gpu-usage">GPU usage</h3>
       
       
       
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+      
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       <li>
-        <a href="https://www.facebook.com/sharer.php?u=https://frankschae.github.io/post/high-weak/&amp;t=High%20weak%20order%20SDE%20solvers" target="_blank" rel="noopener" class="share-btn-facebook">
+        <a href="https://www.facebook.com/sharer.php?u=https%3A%2F%2Ffrankschae.github.io%2Fpost%2Fhigh-weak%2F&amp;t=High&#43;weak&#43;order&#43;SDE&#43;solvers" target="_blank" rel="noopener" class="share-btn-facebook" aria-label="facebook">
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@@ -1137,8 +1298,14 @@ <h3 id="gpu-usage">GPU usage</h3>
       
       
       
+      
+      
+      
+      
+        
+      
       <li>
-        <a href="mailto:?subject=High%20weak%20order%20SDE%20solvers&amp;body=https://frankschae.github.io/post/high-weak/" target="_blank" rel="noopener" class="share-btn-email">
+        <a href="mailto:?subject=High%20weak%20order%20SDE%20solvers&amp;body=https%3A%2F%2Ffrankschae.github.io%2Fpost%2Fhigh-weak%2F" target="_blank" rel="noopener" class="share-btn-email" aria-label="envelope">
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@@ -1150,8 +1317,12 @@ <h3 id="gpu-usage">GPU usage</h3>
       
       
       
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         </button>
       </div>
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-        <pre><code class="tex hljs"></code></pre>
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-<!DOCTYPE html><html lang="en-us" >
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-  
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@@ -654,21 +853,11 @@ <h1>Neural Hybrid Differential Equations</h1>
   <div class="article-container">
 
     <div class="article-style">
-      <p>I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors 
-<a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, 
-<a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>,  
-<a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a>, and 
-<a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a>. This year&rsquo;s project is embedded within the 
-<a href="https://summerofcode.withgoogle.com/organizations/5765643267211264/" target="_blank" rel="noopener">NumFocus</a>/
-<a href="https://sciml.ai" target="_blank" rel="noopener">SciML</a> organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.</p>
+      <p>I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors <a href="https://github.com/ChrisRackauckas" target="_blank" rel="noopener">Chris Rackauckas</a>, <a href="https://github.com/mschauer" target="_blank" rel="noopener">Moritz Schauer</a>,  <a href="https://github.com/YingboMa" target="_blank" rel="noopener">Yingbo Ma</a>, and <a href="https://github.com/mohamed82008" target="_blank" rel="noopener">Mohamed Tarek</a>. This year&rsquo;s project is embedded within the <a href="https://summerofcode.withgoogle.com/organizations/5765643267211264/" target="_blank" rel="noopener">NumFocus</a>/<a href="https://sciml.ai" target="_blank" rel="noopener">SciML</a> organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.</p>
 <p>This first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).</p>
-<p>Note: Please check the 
-<a href="https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/" target="_blank" rel="noopener">SciMLSensitivity.jl docs</a> for a maintained neural hybrid DE tutorial!</p>
+<p>Note: Please check the <a href="https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/" target="_blank" rel="noopener">SciMLSensitivity.jl docs</a> for a maintained neural hybrid DE tutorial!</p>
 <h2 id="hybrid-differential-equations">Hybrid Differential Equations</h2>
-<p>DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by 
-<a href="https://diffeq.sciml.ai/stable/features/callback_functions/" target="_blank" rel="noopener">callbacks</a>. Evidently, the incorporation of discontinuities allows a user to specify changes (<em>events</em>) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by 
-<a href="https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples" target="_blank" rel="noopener">DiscreteCallbacks</a>, implicit events have to be specified by 
-<a href="https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples" target="_blank" rel="noopener">ContinuousCallbacks</a>. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to <code>0</code>. Thus, implicit events require some sort of rootfinding procedure.</p>
+<p>DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by <a href="https://diffeq.sciml.ai/stable/features/callback_functions/" target="_blank" rel="noopener">callbacks</a>. Evidently, the incorporation of discontinuities allows a user to specify changes (<em>events</em>) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by <a href="https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples" target="_blank" rel="noopener">DiscreteCallbacks</a>, implicit events have to be specified by <a href="https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples" target="_blank" rel="noopener">ContinuousCallbacks</a>. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to <code>0</code>. Thus, implicit events require some sort of rootfinding procedure.</p>
 <p>Some relevant examples for hybrid DEs with discrete or continuous callbacks are:</p>
 <ul>
 <li>quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> (<code>ContinuousCallback</code>).</li>
@@ -678,7 +867,7 @@ <h2 id="hybrid-differential-equations">Hybrid Differential Equations</h2>
 <li>pharmacokinetic models<sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>, where explicit dosing times change the drug concentration in the blood (<code>DiscreteCallback</code>). The simplest possible example being the one-compartment model.</li>
 <li>kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (<code>DiscreteCallback</code>).</li>
 </ul>
-<p>The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. <sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> and <sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>.</p>
+<p>The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. <sup id="fnref1:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> and <sup id="fnref1:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>.</p>
 <h2 id="kicked-harmonic-oscillator">Kicked harmonic oscillator</h2>
 <p>Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form</p>
 <p>$$
@@ -687,43 +876,39 @@ <h2 id="kicked-harmonic-oscillator">Kicked harmonic oscillator</h2>
 <p>where $a=0.1$ and $b=1$ with initial conditions</p>
 <p>$$
 \begin{aligned}
-x(t=0) &amp;= 1  \\<br>
+x(t=0) &amp;= 1  \\
 v(t=0) &amp;= \dot{x}(t=0) = 0.
 \end{aligned}<br>
 $$</p>
-<p>This second-order ODE can be 
-<a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order" target="_blank" rel="noopener">reduced</a> to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the <code>DifferentialEquations.jl</code> package. (Instead of doing this reduction manually, we could also use 
-<a href="https://mtk.sciml.ai/stable/tutorials/higher_order/" target="_blank" rel="noopener"><code>ModelingToolkit.jl</code></a> to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a <code>SecondOrderODEProblem</code> implemented.) The Julia code reads:</p>
-<pre><code class="language-julia">using DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity
-using Zygote
-using Random
-u0 = Float32[1.; 0.]
-
-tspan = (0.0f0,50.0f0)
-
-dtsave = 0.5f0
-t = tspan[1]:dtsave:tspan[2]
-
-function oscillator!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  return nothing
-end
+<p>This second-order ODE can be <a href="https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order" target="_blank" rel="noopener">reduced</a> to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the <code>DifferentialEquations.jl</code> package. (Instead of doing this reduction manually, we could also use <a href="https://mtk.sciml.ai/stable/tutorials/higher_order/" target="_blank" rel="noopener"><code>ModelingToolkit.jl</code></a> to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a <code>SecondOrderODEProblem</code> implemented.) The Julia code reads:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">using</span> <span class="n">DiffEqFlux</span><span class="p">,</span> <span class="n">DifferentialEquations</span><span class="p">,</span> <span class="n">Flux</span><span class="p">,</span> <span class="n">Optim</span><span class="p">,</span> <span class="n">Plots</span><span class="p">,</span> <span class="n">DiffEqSensitivity</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Zygote</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Random</span>
+</span></span><span class="line"><span class="cl"><span class="n">u0</span> <span class="o">=</span> <span class="kt">Float32</span><span class="p">[</span><span class="mf">1.</span><span class="p">;</span> <span class="mf">0.</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">tspan</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.0f0</span><span class="p">,</span><span class="mf">50.0f0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">dtsave</span> <span class="o">=</span> <span class="mf">0.5f0</span>
+</span></span><span class="line"><span class="cl"><span class="n">t</span> <span class="o">=</span> <span class="n">tspan</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">:</span><span class="n">dtsave</span><span class="o">:</span><span class="n">tspan</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">oscillator!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="o">//</span><span class="mi">10</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">prob_data</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">oscillator!</span><span class="p">,</span><span class="n">u0</span><span class="p">,</span><span class="n">tspan</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># ODE without kicks</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">prob_data</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">saveat</span><span class="o">=</span><span class="n">t</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="p">[</span><span class="s">&#34;x(t)&#34;</span> <span class="s">&#34;v(t)&#34;</span><span class="p">])</span>
+</span></span></code></pre></div>
 
-prob_data = ODEProblem(oscillator!,u0,tspan)
 
-# ODE without kicks
-pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&quot;x(t)&quot; &quot;v(t)&quot;])
-</code></pre>
 
 
 
 
 
-  
-
-
-
 
 
 
@@ -732,47 +917,37 @@ <h2 id="kicked-harmonic-oscillator">Kicked harmonic oscillator</h2>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/forward_damped_oscillator_no_kicks.png" >
-
-
-  <img src="/img/forward_damped_oscillator_no_kicks.png" alt=""  >
-</a>
-
-
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="forward_damped_oscillator_no_kicks.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as <code>1</code>.</p>
-<pre><code class="language-julia">kicktimes = tspan[1]:1:tspan[2]
-function kick!(integrator)
-  integrator.u[end] += one(eltype(integrator.u))
-end
-cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false))
-
-sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t)
-t_data = sol_data.t
-ode_data = Array(sol_data)
-
-# visualize data
-pl1 = plot(t_data,ode_data[1,:],label=&quot;data x(t)&quot;)
-plot!(pl1,t_data,ode_data[2,:],label=&quot;data v(t)&quot;)
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="n">kicktimes</span> <span class="o">=</span> <span class="n">tspan</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">:</span><span class="mi">1</span><span class="o">:</span><span class="n">tspan</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">kick!</span><span class="p">(</span><span class="n">integrator</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">integrator</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="k">end</span><span class="p">]</span> <span class="o">+=</span> <span class="n">one</span><span class="p">(</span><span class="n">eltype</span><span class="p">(</span><span class="n">integrator</span><span class="o">.</span><span class="n">u</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="n">cb_</span> <span class="o">=</span> <span class="n">PresetTimeCallback</span><span class="p">(</span><span class="n">kicktimes</span><span class="p">,</span><span class="n">kick!</span><span class="p">,</span><span class="n">save_positions</span><span class="o">=</span><span class="p">(</span><span class="nb">false</span><span class="p">,</span><span class="nb">false</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">sol_data</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob_data</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">callback</span><span class="o">=</span><span class="n">cb_</span><span class="p">,</span><span class="n">saveat</span><span class="o">=</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">t_data</span> <span class="o">=</span> <span class="n">sol_data</span><span class="o">.</span><span class="n">t</span>
+</span></span><span class="line"><span class="cl"><span class="n">ode_data</span> <span class="o">=</span> <span class="kt">Array</span><span class="p">(</span><span class="n">sol_data</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># visualize data</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">t_data</span><span class="p">,</span><span class="n">ode_data</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="o">:</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s">&#34;data x(t)&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">plot!</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span><span class="n">t_data</span><span class="p">,</span><span class="n">ode_data</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="o">:</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s">&#34;data v(t)&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">pl2</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">t_data</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">20</span><span class="p">],</span><span class="n">ode_data</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="o">:</span><span class="mi">20</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s">&#34;data x(t)&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">plot!</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span><span class="n">t_data</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">20</span><span class="p">],</span><span class="n">ode_data</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="o">:</span><span class="mi">20</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s">&#34;data v(t)&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span> <span class="n">pl1</span><span class="p">,</span> <span class="n">layout</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">xlabel</span><span class="o">=</span><span class="s">&#34;t&#34;</span><span class="p">)</span>
+</span></span></code></pre></div><p>
 
-pl2 = plot(t_data[1:20],ode_data[1,1:20],label=&quot;data x(t)&quot;)
-plot!(pl2,t_data[1:20],ode_data[2,1:20],label=&quot;data v(t)&quot;)
-pl = plot(pl2, pl1, layout=(1,2), xlabel=&quot;t&quot;)
-</code></pre>
-<p>
 
 
 
 
-  
-
-
-
 
 
 
@@ -781,63 +956,52 @@ <h2 id="kicked-harmonic-oscillator">Kicked harmonic oscillator</h2>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/forward_damped_oscillator.png" >
 
 
-  <img src="/img/forward_damped_oscillator.png" alt=""  >
-</a>
-
-
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="forward_damped_oscillator.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting <code>save_positions=(true,true)</code>, the kicks would be saved before <strong>and</strong> after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations<sup id="fnref:6"><a href="#fn:6" class="footnote-ref" role="doc-noteref">6</a></sup>, we now aim at learning (potentially) missing parts of the model from these data traces.</p>
 <h3 id="high-domain-knowledge">High domain knowledge</h3>
 <p>For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the <code>affect!</code> function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of <code>integrator.u</code>. If we further set the initial value of that component to <code>1</code>, then the neural network only has to learn that <code>du[3]</code> is <code>0</code>. In other words, the output of the neural network must be <code>0</code> for all states <code>u</code>.</p>
-<pre><code class="language-julia">Random.seed!(123)
-nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1))
-p_nn1 = initial_params(nn1)
-
-function f1!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3] = nn1(u[1:2], p)[1]
-  return nothing
-end
-
-affect!(integrator) = integrator.u[2] += integrator.u[3]
-cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false))
-z0 = Float32[u0;one(u0[1])]
-prob1 = ODEProblem(f1!,z0,tspan,p_nn1)
-</code></pre>
-<p>We can easily compare the time evolution of the neural hybrid DE with respect to the data:</p>
-<pre><code class="language-julia"># to visualize the predictions of the trained neural network below
-function visualize(prob,p)
-  _prob = remake(prob,p=p)
-  ode_pred = Array(solve(_prob,Tsit5(),callback=cb,
-                 saveat=dtsave))[1:2,:]
-  pl1 = plot(t_data,ode_pred[1,:],label=&quot;x(t)&quot;)
-  scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=&quot;data x(t)&quot;)
-  pl2 = plot(t_data,ode_pred[2,:],label=&quot;v(t)&quot;)
-  scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=&quot;data v(t)&quot;)
-
-  pl = plot(pl1, pl2, layout=(1,2), xlabel=&quot;t&quot;)
-  return pl, sum(abs2,ode_data .- ode_pred)
-end
-
-pl = plot(solve(prob1,Tsit5(),saveat=t,
-  callback=cb
-  ),label=[&quot;x(t)&quot; &quot;v(t)&quot; &quot;u3(t)&quot;])
-</code></pre>
-
-
-
-
-
-  
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="n">Random</span><span class="o">.</span><span class="n">seed!</span><span class="p">(</span><span class="mi">123</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">nn1</span> <span class="o">=</span> <span class="n">FastChain</span><span class="p">(</span><span class="n">FastDense</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">64</span><span class="p">,</span> <span class="n">tanh</span><span class="p">),</span><span class="n">FastDense</span><span class="p">(</span><span class="mi">64</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">p_nn1</span> <span class="o">=</span> <span class="n">initial_params</span><span class="p">(</span><span class="n">nn1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f1!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="o">//</span><span class="mi">10</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">nn1</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">2</span><span class="p">],</span> <span class="n">p</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">affect!</span><span class="p">(</span><span class="n">integrator</span><span class="p">)</span> <span class="o">=</span> <span class="n">integrator</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">+=</span> <span class="n">integrator</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">cb</span> <span class="o">=</span> <span class="n">PresetTimeCallback</span><span class="p">(</span><span class="n">kicktimes</span><span class="p">,</span><span class="n">affect!</span><span class="p">,</span><span class="n">save_positions</span><span class="o">=</span><span class="p">(</span><span class="nb">false</span><span class="p">,</span><span class="nb">false</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">z0</span> <span class="o">=</span> <span class="kt">Float32</span><span class="p">[</span><span class="n">u0</span><span class="p">;</span><span class="n">one</span><span class="p">(</span><span class="n">u0</span><span class="p">[</span><span class="mi">1</span><span class="p">])]</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob1</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">f1!</span><span class="p">,</span><span class="n">z0</span><span class="p">,</span><span class="n">tspan</span><span class="p">,</span><span class="n">p_nn1</span><span class="p">)</span>
+</span></span></code></pre></div><p>We can easily compare the time evolution of the neural hybrid DE with respect to the data:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># to visualize the predictions of the trained neural network below</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">visualize</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">_prob</span> <span class="o">=</span> <span class="n">remake</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">ode_pred</span> <span class="o">=</span> <span class="kt">Array</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">_prob</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">callback</span><span class="o">=</span><span class="n">cb</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">                 <span class="n">saveat</span><span class="o">=</span><span class="n">dtsave</span><span class="p">))[</span><span class="mi">1</span><span class="o">:</span><span class="mi">2</span><span class="p">,</span><span class="o">:</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">pl1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">t_data</span><span class="p">,</span><span class="n">ode_pred</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="o">:</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s">&#34;x(t)&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">scatter!</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span><span class="n">t_data</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">5</span><span class="o">:</span><span class="k">end</span><span class="p">],</span><span class="n">ode_data</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="o">:</span><span class="mi">5</span><span class="o">:</span><span class="k">end</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s">&#34;data x(t)&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">pl2</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">t_data</span><span class="p">,</span><span class="n">ode_pred</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="o">:</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s">&#34;v(t)&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">scatter!</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span><span class="n">t_data</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">5</span><span class="o">:</span><span class="k">end</span><span class="p">],</span><span class="n">ode_data</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="o">:</span><span class="mi">5</span><span class="o">:</span><span class="k">end</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s">&#34;data v(t)&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span> <span class="n">pl2</span><span class="p">,</span> <span class="n">layout</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span> <span class="n">xlabel</span><span class="o">=</span><span class="s">&#34;t&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="n">pl</span><span class="p">,</span> <span class="n">sum</span><span class="p">(</span><span class="n">abs2</span><span class="p">,</span><span class="n">ode_data</span> <span class="o">.-</span> <span class="n">ode_pred</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">prob1</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">saveat</span><span class="o">=</span><span class="n">t</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">callback</span><span class="o">=</span><span class="n">cb</span>
+</span></span><span class="line"><span class="cl">  <span class="p">),</span><span class="n">label</span><span class="o">=</span><span class="p">[</span><span class="s">&#34;x(t)&#34;</span> <span class="s">&#34;v(t)&#34;</span> <span class="s">&#34;u3(t)&#34;</span><span class="p">])</span>
+</span></span></code></pre></div>
 
 
 
@@ -849,79 +1013,72 @@ <h3 id="high-domain-knowledge">High domain knowledge</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/untrained_nn.png" >
 
 
-  <img src="/img/untrained_nn.png" alt=""  >
-</a>
 
 
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="untrained_nn.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>which (of course) doesn&rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:</p>
-<pre><code class="language-julia">### loss function
-function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint())
-  _prob = remake(prob,p=p)
-  pred = Array(solve(_prob,Tsit5(),callback=cb,
-               saveat=dtsave,sensealg=sensealg))[1:2,:]
-  sum(abs2,ode_data .- pred)
-end
-
-loss(p_nn1)
-</code></pre>
-<p>The recently implemented tools are deeply hidden within the 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl" target="_blank" rel="noopener">DiffEqSensitivity.jl</a> package. However, while the user could previously only choose discrete sensitivities such as <code>ReverseDiffAdjoint()</code> or  <code>ForwardDiffAdjoint()</code> that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as <code>BacksolveAdjoint()</code>,  <code>InterpolatingAdjoint()</code>, and <code>QuadratureAdjoint()</code> as the <code>sensealg</code> for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., 
-<a href="https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=1" target="_blank" rel="noopener">Chris' talk</a> at the SciML symposium at SIAM CSE.</p>
-<pre><code class="language-julia">###################################
-# training loop
-# optimize the parameters for a few epochs with ADAM
-function train(prob, p_nn; sensealg=BacksolveAdjoint())
-  opt = ADAM(0.0003f0)
-  list_plots = []
-  losses = []
-  for epoch in 1:200
-    println(&quot;epoch: $epoch / 200&quot;)
-    _dy, back = Zygote.pullback(p -&gt; loss(p,
-      prob=prob,
-      sensealg=sensealg), p_nn)
-    gs = @time back(one(_dy))[1]
-    push!(losses, _dy)
-    if epoch % 10 == 0
-      # plot every xth epoch
-      pl, test_loss = visualize(prob, p_nn)
-      println(&quot;Loss (epoch: $epoch): $test_loss&quot;)
-      display(pl)
-      push!(list_plots, pl)
-    end
-    Flux.Optimise.update!(opt, p_nn, gs)
-    println(&quot;&quot;)
-  end
-  return losses, list_plots
-end
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c">### loss function</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">loss</span><span class="p">(</span><span class="n">p</span><span class="p">;</span> <span class="n">prob</span><span class="o">=</span><span class="n">prob1</span><span class="p">,</span> <span class="n">sensealg</span> <span class="o">=</span> <span class="n">ReverseDiffAdjoint</span><span class="p">())</span>
+</span></span><span class="line"><span class="cl">  <span class="n">_prob</span> <span class="o">=</span> <span class="n">remake</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">pred</span> <span class="o">=</span> <span class="kt">Array</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">_prob</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">callback</span><span class="o">=</span><span class="n">cb</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">               <span class="n">saveat</span><span class="o">=</span><span class="n">dtsave</span><span class="p">,</span><span class="n">sensealg</span><span class="o">=</span><span class="n">sensealg</span><span class="p">))[</span><span class="mi">1</span><span class="o">:</span><span class="mi">2</span><span class="p">,</span><span class="o">:</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">sum</span><span class="p">(</span><span class="n">abs2</span><span class="p">,</span><span class="n">ode_data</span> <span class="o">.-</span> <span class="n">pred</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">loss</span><span class="p">(</span><span class="n">p_nn1</span><span class="p">)</span>
+</span></span></code></pre></div><p>The recently implemented tools are deeply hidden within the <a href="https://github.com/SciML/DiffEqSensitivity.jl" target="_blank" rel="noopener">DiffEqSensitivity.jl</a> package. However, while the user could previously only choose discrete sensitivities such as <code>ReverseDiffAdjoint()</code> or  <code>ForwardDiffAdjoint()</code> that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as <code>BacksolveAdjoint()</code>,  <code>InterpolatingAdjoint()</code>, and <code>QuadratureAdjoint()</code> as the <code>sensealg</code> for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., <a href="https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=1" target="_blank" rel="noopener">Chris&rsquo; talk</a> at the SciML symposium at SIAM CSE.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c">###################################</span>
+</span></span><span class="line"><span class="cl"><span class="c"># training loop</span>
+</span></span><span class="line"><span class="cl"><span class="c"># optimize the parameters for a few epochs with ADAM</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">train</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span> <span class="n">p_nn</span><span class="p">;</span> <span class="n">sensealg</span><span class="o">=</span><span class="n">BacksolveAdjoint</span><span class="p">())</span>
+</span></span><span class="line"><span class="cl">  <span class="n">opt</span> <span class="o">=</span> <span class="n">ADAM</span><span class="p">(</span><span class="mf">0.0003f0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">list_plots</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">losses</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl">  <span class="k">for</span> <span class="n">epoch</span> <span class="k">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">200</span>
+</span></span><span class="line"><span class="cl">    <span class="n">println</span><span class="p">(</span><span class="s">&#34;epoch: </span><span class="si">$epoch</span><span class="s"> / 200&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">_dy</span><span class="p">,</span> <span class="n">back</span> <span class="o">=</span> <span class="n">Zygote</span><span class="o">.</span><span class="n">pullback</span><span class="p">(</span><span class="n">p</span> <span class="o">-&gt;</span> <span class="n">loss</span><span class="p">(</span><span class="n">p</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">      <span class="n">prob</span><span class="o">=</span><span class="n">prob</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">      <span class="n">sensealg</span><span class="o">=</span><span class="n">sensealg</span><span class="p">),</span> <span class="n">p_nn</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">gs</span> <span class="o">=</span> <span class="nd">@time</span> <span class="n">back</span><span class="p">(</span><span class="n">one</span><span class="p">(</span><span class="n">_dy</span><span class="p">))[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">    <span class="n">push!</span><span class="p">(</span><span class="n">losses</span><span class="p">,</span> <span class="n">_dy</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">if</span> <span class="n">epoch</span> <span class="o">%</span> <span class="mi">10</span> <span class="o">==</span> <span class="mi">0</span>
+</span></span><span class="line"><span class="cl">      <span class="c"># plot every xth epoch</span>
+</span></span><span class="line"><span class="cl">      <span class="n">pl</span><span class="p">,</span> <span class="n">test_loss</span> <span class="o">=</span> <span class="n">visualize</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span> <span class="n">p_nn</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">      <span class="n">println</span><span class="p">(</span><span class="s">&#34;Loss (epoch: </span><span class="si">$epoch</span><span class="s">): </span><span class="si">$test_loss</span><span class="s">&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">      <span class="n">display</span><span class="p">(</span><span class="n">pl</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">      <span class="n">push!</span><span class="p">(</span><span class="n">list_plots</span><span class="p">,</span> <span class="n">pl</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="k">end</span>
+</span></span><span class="line"><span class="cl">    <span class="n">Flux</span><span class="o">.</span><span class="n">Optimise</span><span class="o">.</span><span class="n">update!</span><span class="p">(</span><span class="n">opt</span><span class="p">,</span> <span class="n">p_nn</span><span class="p">,</span> <span class="n">gs</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">println</span><span class="p">(</span><span class="s">&#34;&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="k">end</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="n">losses</span><span class="p">,</span> <span class="n">list_plots</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># plot training loss</span>
+</span></span><span class="line"><span class="cl"><span class="n">losses</span><span class="p">,</span> <span class="n">list_plots</span> <span class="o">=</span> <span class="n">train</span><span class="p">(</span><span class="n">prob1</span><span class="p">,</span> <span class="n">p_nn1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">losses</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mf">1.5</span><span class="p">,</span> <span class="n">xlabel</span> <span class="o">=</span> <span class="s">&#34;epoch&#34;</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="s">&#34;loss&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="nb">false</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl2</span> <span class="o">=</span> <span class="n">list_plots</span><span class="p">[</span><span class="k">end</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl3</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">solve</span><span class="p">(</span><span class="n">prob1</span><span class="p">,</span><span class="n">p</span><span class="o">=</span><span class="n">p_nn1</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">saveat</span><span class="o">=</span><span class="n">t</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">   <span class="n">callback</span><span class="o">=</span><span class="n">cb</span>
+</span></span><span class="line"><span class="cl">  <span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="p">[</span><span class="s">&#34;x(t)&#34;</span> <span class="s">&#34;v(t)&#34;</span> <span class="s">&#34;u3(t)&#34;</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">pl</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span><span class="n">pl3</span><span class="p">)</span>
+</span></span></code></pre></div>
 
-# plot training loss
-losses, list_plots = train(prob1, p_nn1)
-pl1 = plot(losses, lw = 1.5, xlabel = &quot;epoch&quot;, ylabel=&quot;loss&quot;, legend=false)
-pl2 = list_plots[end]
-pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t,
-   callback=cb
-  ), label=[&quot;x(t)&quot; &quot;v(t)&quot; &quot;u3(t)&quot;])
 
-pl = plot(pl2,pl3)
-</code></pre>
 
 
 
 
 
-  
-
-
-
 
 
 
@@ -930,98 +1087,86 @@ <h3 id="high-domain-knowledge">High domain knowledge</h3>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/trained1.png" >
-
-
-  <img src="/img/trained1.png" alt=""  >
-</a>
-
-
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="trained1.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>We see the expected constant value of <code>u[3]</code>, indicating a kick to the velocity of <code>+=1</code>, at the kicking times over the full time interval.</p>
 <h2 id="reducing-the-domain-knowledge">Reducing the domain knowledge</h2>
-<p>If less physical information is included in the model design, the training becomes more difficult, e.g., due to 
-<a href="https://diffeqflux.sciml.ai/dev/examples/local_minima/" target="_blank" rel="noopener">local minima</a>. Possible modification for the kicked oscillator could be</p>
+<p>If less physical information is included in the model design, the training becomes more difficult, e.g., due to <a href="https://diffeqflux.sciml.ai/dev/examples/local_minima/" target="_blank" rel="noopener">local minima</a>. Possible modification for the kicked oscillator could be</p>
 <ul>
 <li>changing the initial condition of the third component of <code>u</code>,</li>
 <li>using another affect function <code>affect!(integrator) = integrator.u[2] = integrator.u[3]</code>,</li>
 <li>dropping the knowledge that only <code>u[2]</code> gets a kick by using a neural network with <code>2</code> outputs (+ a fourth component in the ODE):</li>
 </ul>
-<pre><code class="language-julia">affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4]
-function f2!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-</code></pre>
-<ul>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="n">affect2!</span><span class="p">(</span><span class="n">integrator</span><span class="p">)</span> <span class="o">=</span> <span class="n">integrator</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">integrator</span><span class="o">.</span><span class="n">u</span><span class="p">[</span><span class="mi">3</span><span class="o">:</span><span class="mi">4</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f2!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="o">//</span><span class="mi">10</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">3</span><span class="o">:</span><span class="mi">4</span><span class="p">]</span> <span class="o">.=</span> <span class="n">nn2</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">2</span><span class="p">],</span> <span class="n">nn_weights</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><ul>
 <li>fitting the parameters $a$ and $b$ simultaneously:</li>
 </ul>
-<pre><code class="language-julia">function f3!(du,u,p,t)
-  a = p[end-1]
-  b = p[end]
-  nn_weights = p[1:end-2]
-
-  du[1] = u[2]
-  du[2] = -b*u[1] - a*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-</code></pre>
-<ul>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f3!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">a</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="k">end</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">b</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="k">end</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">nn_weights</span> <span class="o">=</span> <span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="k">end</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="n">b</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">a</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">3</span><span class="o">:</span><span class="mi">4</span><span class="p">]</span> <span class="o">.=</span> <span class="n">nn2</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">2</span><span class="p">],</span> <span class="n">nn_weights</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><ul>
 <li>inferring the entire underlying dynamics using a neural network with <code>4</code> outputs:</li>
 </ul>
-<pre><code class="language-julia">function f4!(du,u,p,t)
-  Ω = nn3(u[1:2], p)
-
-  du[1] = Ω[1]
-  du[2] = Ω[2]
-  du[3:4] .= Ω[3:4]
-  return nothing
-end
-</code></pre>
-<ul>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">f4!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">Ω</span> <span class="o">=</span> <span class="n">nn3</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="o">:</span><span class="mi">2</span><span class="p">],</span> <span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ω</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ω</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">3</span><span class="o">:</span><span class="mi">4</span><span class="p">]</span> <span class="o">.=</span> <span class="n">Ω</span><span class="p">[</span><span class="mi">3</span><span class="o">:</span><span class="mi">4</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="k">return</span> <span class="nb">nothing</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span></code></pre></div><ul>
 <li>etc.</li>
 </ul>
 <h2 id="outlook">Outlook</h2>
 <p>With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to</p>
 <ul>
 <li>refine the adjoints in case of implicit discontinuities (<code>ContinuousCallbacks</code>) and</li>
-<li>support direct usage through the 
-<a href="https://diffeq.sciml.ai/stable/types/jump_types/" target="_blank" rel="noopener">jump problem</a> interface</li>
+<li>support direct usage through the <a href="https://diffeq.sciml.ai/stable/types/jump_types/" target="_blank" rel="noopener">jump problem</a> interface</li>
 </ul>
-<p>in the upcoming weeks. For further information, the interested reader is encouraged to look at the open 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/issues" target="_blank" rel="noopener">issues</a> in the DiffEqSensitivity.jl package.</p>
+<p>in the upcoming weeks. For further information, the interested reader is encouraged to look at the open <a href="https://github.com/SciML/DiffEqSensitivity.jl/issues" target="_blank" rel="noopener">issues</a> in the DiffEqSensitivity.jl package.</p>
 <p>If you have any questions or comments, please don’t hesitate to contact me!</p>
-<section class="footnotes" role="doc-endnotes">
+<div class="footnotes" role="doc-endnotes">
 <hr>
 <ol>
-<li id="fn:1" role="doc-endnote">
-<p>Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. <strong>2</strong>, 035004 (2021) <a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:1">
+<p>Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. <strong>2</strong>, 035004 (2021)&#160;<a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:2" role="doc-endnote">
-<p>Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). <a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:2">
+<p>Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&#160;<a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:3" role="doc-endnote">
-<p>Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). <a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:3">
+<p>Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&#160;<a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:4" role="doc-endnote">
-<p>Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). <a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:4">
+<p>Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&#160;<a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:5" role="doc-endnote">
-<p>Chris Rackauckas, Yingbo Ma, et al., &ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&rdquo; (2020). <a href="#fnref:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:5">
+<p>Chris Rackauckas, Yingbo Ma, et al., &ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&rdquo; (2020).&#160;<a href="#fnref:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:6" role="doc-endnote">
-<p>Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). <a href="#fnref:6" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:6">
+<p>Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&#160;<a href="#fnref:6" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
 </ol>
-</section>
+</div>
 
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+    </div>
+  </script>
+  
+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
+    <script src="https://cdn.jsdelivr.net/gh/julmot/mark.js@8.11.1/dist/jquery.mark.min.js" integrity="sha512-mhbv5DqBMgrWL+32MmsDOt/OAvqr/cHimk6B8y/bx/xS88MVkYGPiVv2ixKVrkywF2qHplNRUvFsAHUdxZ3Krg==" crossorigin="anonymous"></script>
+  
 
-    
-    
-    <script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/3.2.1/fuse.min.js" integrity="sha256-VzgmKYmhsGNNN4Ph1kMW+BjoYJM2jV5i4IlFoeZA9XI=" crossorigin="anonymous"></script>
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-    
 
-    
-    
 
-    
-    
 
-    
-    
 
-    
-    
-    
-    
-    
-    
-    
-    
-    
-      
-    
-    
-    
-    
-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
-    
 
 
 
@@ -1401,32 +1595,54 @@ <h3>Related</h3>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
-    <a href="https://gohugo.io" target="_blank" rel="noopener">Hugo</a>.
 
-    
-    <span class="float-right" aria-hidden="true">
-      <a href="#" class="back-to-top">
-        <span class="button_icon">
-          <i class="fas fa-chevron-up fa-2x"></i>
-        </span>
-      </a>
-    </span>
-    
-  </p>
-</footer>
 
-  </div>
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+  
   
 
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
+
   
 <div id="modal" class="modal fade" role="dialog">
   <div class="modal-dialog">
@@ -1438,7 +1654,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -1453,5 +1670,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
+
+
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+
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+
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+
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+
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+
 </body>
 </html>
diff --git a/post/index.html b/post/index.html
index f00f2c3..8960610 100644
--- a/post/index.html
+++ b/post/index.html
@@ -1,92 +1,107 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/post/">
+
+
+
+
+
+
+
+
 
   
+
+
+
+
+
 
 
   
   
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
   
-  <meta name="theme-color" content="#2962ff">
+
+  
+  
+  
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
-  <script src="/js/mathjax-config.js"></script>
   
 
   
+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
+
   
   
   
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/academicons/1.8.6/css/academicons.min.css" integrity="sha256-uFVgMKfistnJAfoCUQigIl+JfUaP47GrRKjf6CTPVmw=" crossorigin="anonymous">
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-
     
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1.9.4/css/academicons.min.css" integrity="sha512-IW0nhlW5MgNydsXJO40En2EoCkTTjZhI3yuODrZIc8cQ4h1XcF53PsqDHa09NqnkXuIe0Oiyyj171BqZFwISBw==" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
+
     
     
-      
     
     
       
       
-        
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/github.min.css" crossorigin="anonymous" title="hl-light">
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/dracula.min.css" crossorigin="anonymous" title="hl-dark" disabled>
-        
-      
     
-
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.css" integrity="sha256-SHMGCYmST46SoyGgo4YR/9AlK1vf3ff84Aq9yK4hdqM=" crossorigin="anonymous">
     
 
     
-
     
     
-      
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
-      
-      
+    
 
-      
+    
     
       
+      
 
       
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +141,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
   
+  
+  
+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
 
   
   
   
+
+  
+  
+  
+  
+  
+  
+  
+    
+    
+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
   
-  <link rel="stylesheet" href="/css/academic.css">
 
   
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 
 
 
 
-  
 
 
   
   
-
   
-  <link rel="alternate" href="/post/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
+
+
+
+
+
   
 
-  <link rel="manifest" href="/index.webmanifest">
-  <link rel="icon" type="image/png" href="/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_2.png">
-  <link rel="apple-touch-icon" type="image/png" href="/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_192x192_fill_lanczos_center_2.png">
+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
+
+
+
+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/post/" />
+<link rel="canonical" href="https://frankschae.github.io/post/" />
+
+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
+
+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
+<link rel="apple-touch-icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_180x180_fill_lanczos_center_3.png" />
+
+<meta name="theme-color" content="#1565c0" />
+
+
+
+
+
+
+
+
 
-  <link rel="canonical" href="https://frankschae.github.io/post/">
 
   
   
+
+
+
+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
+<meta property="twitter:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" />
+
+
+
   
+
+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/post/" />
+<meta property="og:title" content="Posts | FS" />
+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
+
   
+    <meta property="og:updated_time" content="2021-08-13T21:41:45&#43;02:00" />
   
+
+
+
+
+
+
+
+
+
+
   
   
-    
-    
   
+
   
-  <meta property="twitter:card" content="summary">
   
-  <meta property="og:site_name" content="FS">
-  <meta property="og:url" content="https://frankschae.github.io/post/">
-  <meta property="og:title" content="Posts | FS">
-  <meta property="og:description" content="Postdoctoral researcher"><meta property="og:image" content="https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png">
-  <meta property="twitter:image" content="https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png"><meta property="og:locale" content="en-us">
+    <link rel="alternate" href="/post/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2021-08-13T21:41:45&#43;02:00">
+
   
 
+
   
+  <title>Posts | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>Posts | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   " data-wc-page-id="6370009740e1eb2ee0b7974f10ecc91f" >
+
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
+
+  
+
+
+<aside class="search-modal" id="search">
   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
           <h1>Search</h1>
         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
+  
   
 
 
@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    <button type="button" class="navbar-toggler" data-toggle="collapse"
-            data-target="#navbar-content" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
-    <span><i class="fas fa-bars"></i></span>
-    </button>
-    
 
-    
-    <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    
-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
-        
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
 
-        
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
-        
-        
+        <ul class="navbar-nav d-md-inline-flex">
+          
+
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
-        
-        
-        
           
-        
+          
 
-        
-        
-        
-        
-        
-        
           
+
           
           
           
-        
+            
+          
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#software"><span>Open Source Software</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
+          
+          
+
+          
+
+          
+          
+          
+            
+          
+
+          
+
+          
+          
+          
+          
+
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
 
         
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
@@ -461,65 +681,51 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
+
         
         
-          
-          
-          
-            
-          
-          
         
-
-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
+            </a>
+          </div>
         </li>
+        
 
         
         
 
-      
-
-        
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-    </div>
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+    </div>
+  </nav>
+</header>
 
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+  <div class="page-body">
+    
+    
+    
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+    
 
-  
 
 
 
@@ -556,7 +762,7 @@ <h1>Posts</h1>
   
   
     
-      
+
 
 
 
@@ -572,17 +778,19 @@ <h1>Posts</h1>
   
 
 
-<div class="media stream-item">
+<div class="media stream-item view-compact">
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-    <h3 class="article-title mb-0 mt-0">
+    <div class="section-subheading article-title mb-0 mt-0">
       <a href="/post/gsoc-2021/" >Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis</a>
-    </h3>
+    </div>
 
     
-    <div class="article-style">
-      Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.
-    </div>
+    <a href="/post/gsoc-2021/"  class="summary-link">
+      <div class="article-style">
+        Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -620,6 +828,7 @@ <h3 class="article-title mb-0 mt-0">
   
   
   
+  
 
   
   
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-    <h3 class="article-title mb-0 mt-0">
+    <div class="section-subheading article-title mb-0 mt-0">
       <a href="/post/abstract_differentiation/" >AbstractDifferentiation.jl for AD-backend agnostic code </a>
-    </h3>
+    </div>
 
     
-    <div class="article-style">
-      Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris' article3 for a detailed examination.
-    </div>
+    <a href="/post/abstract_differentiation/"  class="summary-link">
+      <div class="article-style">
+        Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris&rsquo; article3 for a detailed examination.
+      </div>
+    </a>
     
 
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-    <h3 class="article-title mb-0 mt-0">
+    <div class="section-subheading article-title mb-0 mt-0">
       <a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a>
-    </h3>
+    </div>
 
     
-    <div class="article-style">
-      In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
-    </div>
+    <a href="/post/bouncing_ball/"  class="summary-link">
+      <div class="article-style">
+        In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -772,8 +984,8 @@ <h3 class="article-title mb-0 mt-0">
   <div>
     
 
-  
-  <span>Frank Schäfer and Moritz Schauer</span>
+  <span >
+      Frank Schäfer and Moritz Schauer</span>
   </div>
   
   
@@ -800,6 +1012,7 @@ <h3 class="article-title mb-0 mt-0">
   
   
   
+  
 
   
   
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-    <h3 class="article-title mb-0 mt-0">
+    <div class="section-subheading article-title mb-0 mt-0">
       <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
-    </h3>
+    </div>
 
     
-    <div class="article-style">
-      In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
-    </div>
+    <a href="/post/shadowing/"  class="summary-link">
+      <div class="article-style">
+        In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
+      </div>
+    </a>
     
 
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@@ -885,6 +1099,7 @@ <h3 class="article-title mb-0 mt-0">
   
   
   
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-    <h3 class="article-title mb-0 mt-0">
+    <div class="section-subheading article-title mb-0 mt-0">
       <a href="/post/hybridde/" >Neural Hybrid Differential Equations</a>
-    </h3>
+    </div>
 
     
-    <div class="article-style">
-      I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
-    </div>
+    <a href="/post/hybridde/"  class="summary-link">
+      <div class="article-style">
+        I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
+      </div>
+    </a>
     
 
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@@ -970,6 +1186,7 @@ <h3 class="article-title mb-0 mt-0">
   
   
   
+  
 
   
   
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-    <h3 class="article-title mb-0 mt-0">
+    <div class="section-subheading article-title mb-0 mt-0">
       <a href="/post/gsoc-2020/" >High weak order solvers and adjoint sensitivity analysis for stochastic differential equations</a>
-    </h3>
+    </div>
 
     
-    <div class="article-style">
-      Project summary In this project, we have implemented new promising tools within the SciML organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
-    </div>
+    <a href="/post/gsoc-2020/"  class="summary-link">
+      <div class="article-style">
+        Project summary In this project, we have implemented new promising tools within the SciML organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
+      </div>
+    </a>
     
 
     <div class="stream-meta article-metadata">
@@ -1055,6 +1273,7 @@ <h3 class="article-title mb-0 mt-0">
   
   
   
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-    <h3 class="article-title mb-0 mt-0">
+    <div class="section-subheading article-title mb-0 mt-0">
       <a href="/post/high-weak/" >High weak order SDE solvers</a>
-    </h3>
+    </div>
 
     
-    <div class="article-style">
-      This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
+    <a href="/post/high-weak/"  class="summary-link">
+      <div class="article-style">
+        This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
 After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
-    </div>
+      </div>
+    </a>
     
 
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@@ -1143,6 +1363,7 @@ <h3 class="article-title mb-0 mt-0">
   
   
   
+  
 
   
   
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-    
   
     
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-    <h3 class="article-title mb-0 mt-0">
+    <div class="section-subheading article-title mb-0 mt-0">
       <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a>
-    </h3>
+    </div>
 
     
-    <div class="article-style">
-      First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
-    </div>
+    <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/"  class="summary-link">
+      <div class="article-style">
+        First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
+      </div>
+    </a>
     
 
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@@ -1406,7 +1688,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
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-        <pre><code class="tex hljs"></code></pre>
+        
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@@ -1421,5 +1704,25 @@ <h5 class="modal-title">Cite</h5>
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diff --git a/post/index.xml b/post/index.xml
index 7032fdd..74d6275 100644
--- a/post/index.xml
+++ b/post/index.xml
@@ -5,9 +5,9 @@
     <link>https://frankschae.github.io/post/</link>
       <atom:link href="https://frankschae.github.io/post/index.xml" rel="self" type="application/rss+xml" />
     <description>Posts</description>
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+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Fri, 13 Aug 2021 21:41:45 +0200</lastBuildDate>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
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     </image>
@@ -19,128 +19,91 @@
       <guid>https://frankschae.github.io/post/gsoc-2021/</guid>
       <description>&lt;h2 id=&#34;project-summary&#34;&gt;Project summary&lt;/h2&gt;
 &lt;p&gt;In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.&lt;/p&gt;
-&lt;p&gt;Possible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/projects/#5357798591823872&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the GSoC page&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Possible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available &lt;a href=&#34;https://summerofcode.withgoogle.com/projects/#5357798591823872&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the GSoC page&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;blog-posts&#34;&gt;Blog posts&lt;/h2&gt;
 &lt;p&gt;The following blog posts describe the work throughout the GSoC period in more detail:&lt;/p&gt;
 &lt;ol&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/hybridde/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Neural Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/shadowing/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/bouncing_ball/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity Analysis of Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/abstract_differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl for AD-backend agnostic code&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/hybridde/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Neural Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/shadowing/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/bouncing_ball/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity Analysis of Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/abstract_differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl for AD-backend agnostic code&lt;/a&gt;&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;docs&#34;&gt;Docs&lt;/h2&gt;
-&lt;p&gt;Documentation with respect to the adjoint sensitivity tools will be available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the local sensitivity analysis&lt;/a&gt; and 
-&lt;a href=&#34;http://scimlbase.sciml.ai/dev/fundamentals/Differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the control of automatic differentiation choices&lt;/a&gt; pages.&lt;/p&gt;
+&lt;p&gt;Documentation with respect to the adjoint sensitivity tools will be available &lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the local sensitivity analysis&lt;/a&gt; and &lt;a href=&#34;http://scimlbase.sciml.ai/dev/fundamentals/Differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the control of automatic differentiation choices&lt;/a&gt; pages.&lt;/p&gt;
 &lt;h2 id=&#34;achievements&#34;&gt;Achievements&lt;/h2&gt;
 &lt;p&gt;Below is a list of PRs in the various repositories in chronological order.&lt;/p&gt;
 &lt;h4 id=&#34;diffeqsensitivityjl&#34;&gt;DiffEqSensitivity.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/415&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add additive noise downstream test for DiffEqFlux&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/416&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallback fixes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/417&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for changes of p in callbacks&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/418&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix for using the correct uleft/pleft in continuous callback&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/419&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix broadcasting error on steady state adjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/420&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Forward Least Squares Shadowing (LSS)&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/422&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint-mode for the LSS method&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/423&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve dispatch for LSS methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/437&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Non-Intrusive Least Square Shadowing (NILSS)&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/442&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve for NILSS&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/443&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove allocation in NILSS&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/444&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Handle additional callback case&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;State-dependent Continuous Callbacks for BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/474&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;QuadratureAdjoint() for ContinuousCallback&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/475&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;More tests for Neural ODEs with callbacks for different sensitivity algorithms&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/476&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support for PeriodicCallbacks in continuous adjoint methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/415&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add additive noise downstream test for DiffEqFlux&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/416&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallback fixes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/417&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for changes of p in callbacks&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/418&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix for using the correct uleft/pleft in continuous callback&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/419&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix broadcasting error on steady state adjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/420&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Forward Least Squares Shadowing (LSS)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/422&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint-mode for the LSS method&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/423&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve dispatch for LSS methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/437&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Non-Intrusive Least Square Shadowing (NILSS)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/442&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve for NILSS&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/443&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove allocation in NILSS&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/444&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Handle additional callback case&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;State-dependent Continuous Callbacks for BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/474&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;QuadratureAdjoint() for ContinuousCallback&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/475&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;More tests for Neural ODEs with callbacks for different sensitivity algorithms&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/476&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support for PeriodicCallbacks in continuous adjoint methods&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;abstractdifferentiationjl&#34;&gt;AbstractDifferentiation.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/2&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes gradient, Jacobian, Hessian, and vjp tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/2&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes gradient, Jacobian, Hessian, and vjp tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/3&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add ForwardDiff and Zygote&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/3&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add ForwardDiff and Zygote&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;ordinarydiffeqjl&#34;&gt;OrdinaryDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/OrdinaryDiffEq.jl/pull/1424&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix discrete reverse mode for some standard controllers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/OrdinaryDiffEq.jl/pull/1424&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix discrete reverse mode for some standard controllers&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqcallbacksjl&#34;&gt;DiffEqCallbacks.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqCallbacks.jl/pull/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Introduce a PeriodicCallbackAffect struct&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqCallbacks.jl/pull/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Introduce a PeriodicCallbackAffect struct&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;steadystatediffeqjl&#34;&gt;SteadyStateDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/SteadyStateDiffEq.jl/pull/31&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Convert alg.tspan to type of prob.u0&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/SteadyStateDiffEq.jl/pull/31&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Convert alg.tspan to type of prob.u0&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;chainrulesjl&#34;&gt;ChainRules.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/506&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Do not differentiate through the construction of BitArray&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/508&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Use splatting in BitArray&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/506&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Do not differentiate through the construction of BitArray&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/508&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Use splatting in BitArray&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqnoiseprocessjl&#34;&gt;DiffEqNoiseProcess.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/94&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow solvers to use Noise Grid with SVectors&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/94&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow solvers to use Noise Grid with SVectors&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;stochasticdiffeqjl&#34;&gt;StochasticDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/428&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove Ihat2 matrix from weak solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/428&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove Ihat2 matrix from weak solvers&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqdocsjl&#34;&gt;DiffEqDocs.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/490&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Small typo on plot page&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/492&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add docs for shadowing methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/490&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Small typo on plot page&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/492&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add docs for shadowing methods&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;future-work&#34;&gt;Future work&lt;/h2&gt;
 &lt;p&gt;Besides the implementation of more shadowing methods, such as&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/1801.08674&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NILSAS&lt;/a&gt;,&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/1711.06633&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FD-NILSS&lt;/a&gt;, or&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/2009.00595&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fast linear response&lt;/a&gt;,&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/1801.08674&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NILSAS&lt;/a&gt;,&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/1711.06633&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FD-NILSS&lt;/a&gt;, or&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/2009.00595&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fast linear response&lt;/a&gt;,&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;we are planning to&lt;/p&gt;
 &lt;ul&gt;
@@ -148,20 +111,10 @@
 &lt;li&gt;refine the AbstractDifferentiation.jl package and use it within DiffEqSensitivity.jl,&lt;/li&gt;
 &lt;li&gt;add more docs and examples.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the 
-&lt;a href=&#34;https://discourse.julialang.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julia language discourse&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the &lt;a href=&#34;https://discourse.julialang.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julia language discourse&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;acknowledgement&#34;&gt;Acknowledgement&lt;/h2&gt;
-&lt;p&gt;Many thanks to my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times.
-I would also like to thank 
-&lt;a href=&#34;https://quantumtheory-bruder.physik.unibas.ch/en/people/group-members/christoph-bruder/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/arnoldjulian&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julian Arnold&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mako-git&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Martin Koppenhöfer&lt;/a&gt; for helpful comments on my blog posts. Special thanks to 
-&lt;a href=&#34;https://github.com/Zymrael&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Michael Poli&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/massastrello&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stefano Massaroli&lt;/a&gt; for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
+&lt;p&gt;Many thanks to my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times.
+I would also like to thank &lt;a href=&#34;https://quantumtheory-bruder.physik.unibas.ch/en/people/group-members/christoph-bruder/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, &lt;a href=&#34;https://github.com/arnoldjulian&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julian Arnold&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mako-git&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Martin Koppenhöfer&lt;/a&gt; for helpful comments on my blog posts. Special thanks to &lt;a href=&#34;https://github.com/Zymrael&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Michael Poli&lt;/a&gt; and &lt;a href=&#34;https://github.com/massastrello&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stefano Massaroli&lt;/a&gt; for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
 </description>
     </item>
     
@@ -170,33 +123,17 @@ I would also like to thank
       <link>https://frankschae.github.io/post/abstract_differentiation/</link>
       <pubDate>Sun, 01 Aug 2021 12:03:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/abstract_differentiation/</guid>
-      <description>&lt;p&gt;
-&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is 
-&lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&#39; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and 
-&lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, 
-&lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different 
-&lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by 
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a 
-&lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
-&lt;p&gt;
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
+      <description>&lt;p&gt;&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is &lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&amp;rsquo; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and &lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, &lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different &lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by &lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a &lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
+&lt;p&gt;&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
 &lt;h2 id=&#34;optimization-of-the-rosenbrock-function&#34;&gt;Optimization of the Rosenbrock function&lt;/h2&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
+&lt;p&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
 &lt;p&gt;$$
 g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2.
 $$&lt;/p&gt;
 &lt;p&gt;The function $g$ has a global minimum at $(x_1^\star, x_2^\star)= (a, a^2)$ with $g(x_1^\star, x_2^\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.&lt;/p&gt;
-&lt;p&gt;Let us now implement the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
+&lt;p&gt;Let us now implement the &lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
 &lt;p&gt;$$
 S({\bf x}) = \frac{1}{2} \sum_{i=1}^M f_i({\bf x})^2.
 $$&lt;/p&gt;
@@ -204,298 +141,275 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 {\bf x}^{k+1} = {\bf x}^k - \alpha_k \left(J^T J \right)^{-1} J^T f({\bf x}^k),
 $$
-where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a 
-&lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
+where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a &lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;The following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\bf x^\star}=(1,1)$ and our initial guess ${\bf x_0}=(0,-0.1)$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Pkg
-path = @__DIR__
-cd(path); Pkg.activate(&amp;quot;.&amp;quot;); Pkg.instantiate()
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@__DIR__&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;activate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;.&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;instantiate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## AbstractDifferentiation is not released yet!!&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AbstractDifferentiation&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Enzyme&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;UnPack&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Rosenbrock function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualization&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt;    
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.6&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.5&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;linealpha&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;colorbar&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;camera&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;heatmap&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_0&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:circle&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:green&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x^\star&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:star&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:red&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Rosenbrock.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-## AbstractDifferentiation is not released yet!!
 
-using AbstractDifferentiation
-using Test, LinearAlgebra
-using FiniteDifferences, ForwardDiff, Zygote
-using Enzyme, UnPack
-using Plots, LaTeXStrings
-# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6
 
-## Rosenbrock function
-# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²
-g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2
 
-# visualization
-p = [1.0,100.0]
-x₀ = [0.0,-0.1]
-xopt = [1.0,1.0]
 
-do_plot = true
-if do_plot    
-    x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5
-    z = Surface((x₁,x₂)-&amp;gt;g([x₁,x₂],p), x₁, x₂)
-    pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true,
-                labelfontsize=20,camera = (3,50),
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
 
-    pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp),
-                labelfontsize=20,
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
-    scatter!(pl2, [(x₀[1],x₀[2])], label=L&amp;quot;x_0&amp;quot;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green)
-    scatter!(pl2, [(xopt[1],xopt[2])],label=L&amp;quot;x^\star&amp;quot;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red)
 
-    pl = plot(pl1,pl2, layout=(2,1))
-    savefig(pl, &amp;quot;Rosenbrock.png&amp;quot;)
-end
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
 
 
-
-
-
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Rosenbrock.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;To apply the Gauss-Newton algorithm to the Rosenbrock function $g$, we first cast $g$ into an appropriate form fulfilling $S({\bf x})$, i.e., we use:&lt;/p&gt;
 &lt;p&gt;$$
 f:\mathbb{R}^2\rightarrow\mathbb{R}^2:  {\bf x} \mapsto \begin{pmatrix}
-f_1({\bf x}) \\&lt;br&gt;
-f_2({\bf x}) \\&lt;br&gt;
+f_1({\bf x}) \\
+f_2({\bf x}) \\
 \end{pmatrix} = \begin{pmatrix}
-\sqrt{2}(a-x_1) \\&lt;br&gt;
-\sqrt{2b}(x_2-x_1^2)\\&lt;br&gt;
+\sqrt{2}(a-x_1) \\
+\sqrt{2b}(x_2-x_1^2)\\
 \end{pmatrix},
 $$&lt;/p&gt;
 &lt;p&gt;instead of $g$. We can easily compute the Jacobian of $f$ manually&lt;/p&gt;
 &lt;p&gt;$$
 J =  \begin{pmatrix}
--\sqrt{2} &amp;amp; 0 \\&lt;br&gt;
--2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\&lt;br&gt;
+-\sqrt{2} &amp;amp; 0 \\
+-2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\
 \end{pmatrix}.
 $$&lt;/p&gt;
 &lt;p&gt;We can then implement a (simple, non-optimized) version of the Gauss-Newton algorithm as follows.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# bring Rosenbrock function into the form &amp;quot;sum of squares of functions&amp;quot;
-f1(x,p) = convert(eltype(x),sqrt(2))*(p[1]-x[1])
-f2(x,p) = convert(eltype(x),sqrt(2*p[2]))*(x[2]-x[1]^2)
-f(x,p) = [f1(x,p),f2(x,p)]
-function f(res,x,p) # Enzyme works with inplace functions
-	res[1] = f1(x,p)
-	res[2] = f2(x,p)
-	return nothing
-end
-
-## manually pre-defined Jacobian
-function Jacobian(x,p)
-  [-convert(eltype(x),sqrt(2))   0
-  -2*x[1]*convert(eltype(x),sqrt(2*p[2]))  convert(eltype(x),sqrt(2*p[2]))]
-end
-
-## Gauss-Newton scheme
-function GaussNewton!(xs, x, p; maxiter=8, backend=nothing)
-    for i=1:maxiter
-        x = step(x, p, backend)
-        @info i
-        @show x
-        push!(xs, x)
-    end
-    return xs, x
-end
-done(x,x2,p) = g(x2,p) &amp;lt; g(x,p)
-function step(x, p, backend::Nothing, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = Jacobian(x,p)
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;xs = [x₀]
-GaussNewton!(xs, x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# output:
-[ Info: 1 ]
-x = [0.125, -0.08750000000000001]
-[ Info: 2 ]
-x = [0.234375, -0.047265625000000006]
-[ Info: 3 ]
-x = [0.4257812499999995, 0.06800537109374968]
-[ Info: 4 ]
-x = [0.5693359374999986, 0.21857223510742047]
-[ Info: 5 ]
-x = [0.784667968749996, 0.5165503501892037]
-[ Info: 6 ]
-x = [0.9999999999999961, 0.9536321163177449]
-[ Info: 7 ]
-x = [0.9999999999999989, 0.9999999999999999]
-[ Info: 8 ]
-x = [1.0, 1.0]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of 
-&lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## FiniteDifferences
-struct FDMBackend{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend() = FDMBackend(central_fdm(5, 1))
-const fdm_backend = FDMBackend()
-# Minimal interface
-AD.@primitive function jacobian(ab::FDMBackend, f, xs...)
-    return FiniteDifferences.jacobian(ab.alg, f, xs...)
-end
-
-# AD Jacobian returns tuple
-# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]
-# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]
-
-@test AD.jacobian(fdm_backend, x-&amp;gt;f(x,p), x₀)[1] ≈ Jacobian(x₀, p)
-@test AD.jacobian(fdm_backend, f, x₀, p)[1] ≈ Jacobian(x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function step(x, p, backend, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = AD.jacobian(backend, f, x, p)[1]
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-
-
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=fdm_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If we want to use reverse-mode AD instead, for example via 
-&lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Zygote
-struct ZygoteBackend &amp;lt;: AD.AbstractReverseMode end
-const zygote_backend = ZygoteBackend()
-AD.@primitive function pullback_function(ab::ZygoteBackend, f, xs...)
-    return function (vs)
-        # Supports only single output
-        _, back = Zygote.pullback(f, xs...)
-        if vs isa AbstractVector
-            back(vs)
-        else
-            @assert length(vs) == 1
-            back(vs[1])
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(zygote_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=zygote_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (
-&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## ForwardDiff
-struct ForwardDiffBackend &amp;lt;: AD.AbstractForwardMode end
-const forwarddiff_backend = ForwardDiffBackend()
-AD.@primitive function pushforward_function(ab::ForwardDiffBackend, f, xs...)
-    # jvp = f&#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0
-    return function (vs)
-        if xs isa Tuple
-            @assert length(xs) &amp;lt;= 2
-            if length(xs) == 1
-                (ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1]),0),)
-            else
-                ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1], xs[2]+h*vs[2]),0)
-            end
-        else
-            ForwardDiff.derivative(h-&amp;gt;f(xs+h*vs),0)
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(forwarddiff_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=forwarddiff_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;where we have used that the Jacobian-vector product $f&#39;(x)v$, i.e., the primitives of forward-mode AD, can be computed by 
-&lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. 
-&lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Enzyme
-struct EnzymeBackend{T1,T2,T3,T4} &amp;lt;: AD.AbstractReverseMode
-    out::T1
-    λ::T2
-    ∂f_∂x::T3
-    ∂f_∂p::T4
-end
-
-out = zero(x₀)
-λ = zero(x₀)
-∂f_∂x = zero(x₀)
-∂f_∂p = zero(p)
-
-const enzyme_backend = EnzymeBackend(out,λ,∂f_∂x,∂f_∂p)
-AD.@primitive function pullback_function(ab::EnzymeBackend, f, xs...)
-    return function (vs)  
-        # enzyme works only with inplace functions
-        if !(vs isa AbstractVector)
-            @assert length(vs) == 1 # Supports only single output
-            vs = vs[1]
-        end
-
-        if xs isa Tuple
-            @assert length(xs) == 2  # hard-coded for use case with two inputs
-            x₀ = xs[1]
-            p = xs[2]
-        end
-
-        @unpack out, λ, ∂f_∂x, ∂f_∂p = ab # cached in the struct, could also be created in here
-
-        ∂f_∂x .*= false
-        ∂f_∂p .*= false
-        out .*= false
-
-        copyto!(λ, vs)
-
-        autodiff(Duplicated(out, λ), Duplicated(x₀, ∂f_∂x), Duplicated(p, ∂f_∂p)) do _out,_x, _p
-            f(_out,_x,_p)
-        end
-        return (∂f_∂x,∂f_∂p)
-    end
-end
-AD.isinplace(ab::EnzymeBackend) = true
-AD.primalvalue(ab::EnzymeBackend, nothing, f, xs) = (f(ab.out,xs...);return ab.out)
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(enzyme_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=enzyme_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# bring Rosenbrock function into the form &amp;#34;sum of squares of functions&amp;#34;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Enzyme works with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## manually pre-defined Jacobian&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;   &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Gauss-Newton scheme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# output:&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.125&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.08750000000000001&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.234375&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.047265625000000006&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.4257812499999995&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.06800537109374968&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5693359374999986&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.21857223510742047&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.784667968749996&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5165503501892037&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;6&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999961&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9536321163177449&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;7&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999989&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9999999999999999&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of &lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## FiniteDifferences&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Minimal interface&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# AD Jacobian returns tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If we want to use reverse-mode AD instead, for example via &lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## ForwardDiff&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractForwardMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c&#34;&gt;# jvp = f&amp;#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;where we have used that the Jacobian-vector product $f&amp;rsquo;(x)v$, i.e., the primitives of forward-mode AD, can be computed by &lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. &lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Enzyme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;  
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# enzyme works only with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;  &lt;span class=&#34;c&#34;&gt;# hard-coded for use case with two inputs&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@unpack&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# cached in the struct, could also be created in here&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;autodiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;do&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;isinplace&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;primalvalue&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
 &lt;h2 id=&#34;some-current-glitches&#34;&gt;Some current glitches&lt;/h2&gt;
 &lt;p&gt;First, the push forward of a tuple of vectors, e.g., $(v_1, v_2)$, for a function with several input arguments is currently ambiguous. While &lt;code&gt;AD.jacobian&lt;/code&gt; primitives and &lt;code&gt;AD.pullback_function&lt;/code&gt; primitives interpret the push forward of our $f$ function as&lt;/p&gt;
 &lt;p&gt;$$
@@ -505,86 +419,77 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\partial f(x_0,p)}{\partial x} v_1 + \frac{\partial f(x_0,p)}{\partial p} v_2.
 $$&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous
-vs = (randn(2), randn(2))
-res1 = AD.pushforward_function(fdm_backend, f, x₀, p)(vs)
-res2 = AD.pushforward_function(forwarddiff_backend, f, x₀, p)(vs)
-
-@test res2 ≈ res1[1] + res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;res2a = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((vs[1], zero(vs[2])))
-res2b = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((zero(vs[1]), vs[2]))
-
-@test res2a ≈ res1[1]
-@test res2b ≈ res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;struct FDMBackend2{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend2() = FDMBackend2(central_fdm(5, 1))
-const fdm_backend2 = FDMBackend2()
-AD.@primitive function pushforward_function(ab::FDMBackend2, f, xs...)
-    return function (vs)
-        FDM.jvp(ab.alg, f, tuple.(xs, vs)...)
-    end
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian only defined with respect to single input variable
-@test_throws AssertionError H1 = AD.hessian(forwarddiff_backend, g, x₀, p)
-H1 = AD.hessian(forwarddiff_backend, x-&amp;gt;g(x,p), x₀)
-H2 = AD.hessian(forwarddiff_backend, p-&amp;gt;g(x₀,p), p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)
-backends = AD.HigherOrderBackend((forwarddiff_backend,zygote_backend))
-H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;FDM&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jvp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tuple&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian only defined with respect to single input variable&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test_throws&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AssertionError&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;HigherOrderBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;There are many other use cases, e.g.,&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;AbstractDifferentiation.jl is by no means complete yet. We are still in the very early stages, but we hope to make significant progress in the coming weeks. Some of the next steps are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;fixing remaining bugs, e.g., with respect to the computation of the Hessian and&lt;/li&gt;
-&lt;li&gt;adding AD/Finite Differentiation packages such as 
-&lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
+&lt;li&gt;adding AD/Finite Differentiation packages such as &lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -596,69 +501,77 @@ H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
       <description>&lt;p&gt;In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. We highlight differences between explicit&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and implicit events&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;. As a paradigmatic example, we consider a bouncing ball described by the ODE&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\&lt;br&gt;
+\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\
 \text{d}v(t) &amp;amp;= -\mathrm g\thinspace  \text{d}t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;with initial condition&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t=0) &amp;amp;= z_0 = 5, \\&lt;br&gt;
+z(t=0) &amp;amp;= z_0 = 5, \\
 v(t=0) &amp;amp;= v_0 = -0.1.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;The initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.&lt;/p&gt;
-&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
+&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
 &lt;p&gt;When ignoring the bounces, we can straightforwardly integrate the ODE analytically&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\&lt;br&gt;
+z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\
 v(t) &amp;amp;= v_0 - \mathrm g\thinspace  t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;or numerically using the OrdinaryDiffEq package from the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### simulate forward
+&lt;p&gt;or numerically using the OrdinaryDiffEq package from the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### simulate forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# dynamics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# parameters and solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;5.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.9&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward_no_bounce.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
+
+
 
-using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity
-using Plots, LaTeXStrings
 
-# dynamics
-function f(du,u,p,t)
-  du[1] = u[2]
-  du[2] = -p[1]
-end
 
-# parameters and solve
-z0 = 5.0
-v0 = -0.1
-t0 = 0.0
-tend = 1.9
-g = 10
-γ = 0.8
 
-u0 = [z0,v0]
-tspan = (t0,tend)
-p = [g, γ]
-prob = ODEProblem(f,u0,tspan,p)
 
-# plot forward trajectory
-sol = solve(prob,Tsit5(),saveat=0.1)
-pl = plot(sol, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomleft)
-hline!(pl, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34;&gt;&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
+&lt;/p&gt;
 
 
 
@@ -667,25 +580,27 @@ savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+
+
+
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward_no_bounce.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Of course, this way the ball continues to fall through the barrier at $z=0$.&lt;/p&gt;
 &lt;h2 id=&#34;forward-simulation-with-events&#34;&gt;Forward simulation with events&lt;/h2&gt;
 &lt;p&gt;At time $\tau$ around $\tau \approx 1$, the ball hits the ground $z(\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\tau$ with new initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau}) &amp;amp;=  z(\tau-) ,\\&lt;br&gt;
+z({\tau}) &amp;amp;=  z(\tau-) ,\\
 v({\tau})&amp;amp;= -\gamma v(\tau-) ,
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -702,15 +617,15 @@ $$&lt;/p&gt;
 &lt;p&gt;The full explicit trajectory $z_{\rm exp}(t) = z(t)$ is determined by&lt;/p&gt;
 &lt;p&gt;$$
 z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\&lt;br&gt;
--0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\
+-0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\
 \quad  - (-0.99005 + t) (-0.99005 \mathrm g + v_0)\gamma  ,&amp;amp; \forall t \ge \tau,
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;where we used&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\&lt;br&gt;
+z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\
 v({\tau})&amp;amp;= -\gamma v({\tau-}) = -\gamma(v_0 - 0.99005 \mathrm g)  .
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -719,15 +634,14 @@ $$&lt;/p&gt;
 a(z,v) = (z, -\gamma v).
 $$&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;DiscreteCallback&lt;/code&gt; in this case to simulate the system.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with DiscreteCallback (explicit event)
-tstar = (v0 + sqrt(v0^2+2*z0*g))/g
-condition1(u,t,integrator) = (t == tstar)
-affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2]
-cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true))
-
-sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar])
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with DiscreteCallback (explicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiscreteCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstops&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
 &lt;h3 id=&#34;implicit-events&#34;&gt;Implicit events&lt;/h3&gt;
 &lt;p&gt;The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)&lt;/p&gt;
 &lt;p&gt;$$
@@ -736,7 +650,7 @@ $$&lt;/p&gt;
 &lt;p&gt;where an event occurs at time $\tau$ if $g(z(\tau),v(\tau),p,\tau) = 0$. We have already used this condition to define our impact time $\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time  $g(z,v,p,t) = 0$.&lt;/p&gt;
 &lt;p&gt;As in the previous case, we can analytically compute the full trajectory of the ball. By substituting the formula for $\tau$ we have at the event time&lt;/p&gt;
 &lt;p&gt;\begin{aligned}
-z({\tau})&amp;amp;= 0, \\&lt;br&gt;
+z({\tau})&amp;amp;= 0, \\
 v({\tau}-)&amp;amp;= - \sqrt{v_0^2 + 2 \mathrm g z_0}
 \end{aligned}&lt;/p&gt;
 &lt;p&gt;for the left limit and&lt;/p&gt;
@@ -746,33 +660,26 @@ v({\tau})&amp;amp;= \gamma \sqrt{v_0^2 + 2 \mathrm g z_0}
 &lt;p&gt;right after the bounce. Thus, the full trajectory $z_{\rm imp}(t) = z(t)$ is given by&lt;/p&gt;
 &lt;p&gt;$$
 (\star) \quad z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\&lt;br&gt;
--\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\
+-\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\
 \quad\cdot \space (-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0} (1 + 2 \gamma)), &amp;amp; \forall t \ge \tau.
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;This is correct even if one substitutes, e.g., a value with higher precision $\mathrm g = 9.81$ for the gravitation constant.&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;ContinuousCallback&lt;/code&gt; in this case.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with ContinuousCallback (implicit event)
-condition2(u,t,integrator) = u[1] # Event happens when condition2(u,t,integrator) == 0
-cb2 = ContinuousCallback(condition2,affect!,save_positions=(true,true))
-sol2 = solve(prob,Tsit5(),callback=cb2,saveat=0.1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# plot forward trajectory
-pl1 = plot(sol1, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;explicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-pl2 = plot(sol2, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;implicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-hline!(pl1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-hline!(pl2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-pl = plot(pl1,pl2)
-savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with ContinuousCallback (implicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Event happens when condition2(u,t,integrator) == 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ContinuousCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;explicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;implicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -784,18 +691,16 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In addition, the implicitly defined impact time via the &lt;code&gt;ContinuousCallback&lt;/code&gt; also changes appropriately when changing the initial conditions or the parameters, for example when using  $\mathrm g = 9.81$ for the gravitation constant. In other words, the event time $\tau=\tau(p,z_0,v_0,t_0)$ is a function of the parameters and initial conditions, and is implicitly defined by the event condition.&lt;/p&gt;
 &lt;p&gt;Suppose we let the ball drop from a somewhat higher position now. Does an increase in height $z$ at $t=0$ give an increase or decrease in height at the end time $t_\text{end}=1.9$? This is something we can answer with sensitivity analysis. For example if we increase the height by (a fraction of) one unit then using $(\star)$&lt;/p&gt;
@@ -804,36 +709,33 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 $$&lt;/p&gt;
 &lt;p&gt;meaning the height at $t_\text{end}$ is also by a corresponding fraction of 0.84 units higher.&lt;/p&gt;
 &lt;p&gt;We can verify this visually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# animate forward trajectory
-sol3 = solve(remake(prob,u0=[u0[1]+0.5,u0[2]]),Tsit5(),callback=cb2,saveat=0.01)
-
-plt2 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;, xlims=(t0,tend))
-hline!(plt2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-plot!(plt2, sol3, tspan=(t0,tend), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)
-
-list_plots = []
-for t in sol3.t
-  tstart = 0.0
-
-  plt1 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;)
-  hline!(plt1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-  plot!(plt1, sol3, tspan=(t0,t), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-  scatter!(plt1,[t,t], sol3(t), color=[1, 2], label=false)
-  plt = plot(plt1,plt2)
-  push!(list_plots, plt)
-end
-
-plot(list_plots[100])
-
-anim = animate(list_plots,every=1)
-&lt;/code&gt;&lt;/pre&gt;
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# animate forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]]),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
-  
 
 
 
@@ -845,18 +747,13 @@ anim = animate(list_plots,every=1)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB.gif&#34; &gt;
 
-
-  &lt;img src=&#34;https://frankschae.github.io/img/BB.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;The original curve is shown in black in the figure above.&lt;/p&gt;
 &lt;h2 id=&#34;sensitivity-analysis-with-events&#34;&gt;Sensitivity analysis with events&lt;/h2&gt;
@@ -871,7 +768,7 @@ $$&lt;/p&gt;
 &lt;p&gt;Inserting our results for $z(t) = z_{\rm exp}(t)$ instead of $z(t) = z_{\rm imp}(t)$ at $t_{\text{end}}$, a different value for the sensitivity is obtained (a value ignoring the changes in $z$ due to changes in the bouncing time $\tau$), e.g.,&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L^{\text{exp}}}{\text{d} z_0} = \begin{cases}
-1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\&lt;br&gt;
+1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\
 0.84 &amp;amp; \text{with } z(t) = z_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -882,7 +779,7 @@ $$&lt;/p&gt;
 &lt;p&gt;we could be interested in the sensitivity of $L$ with respect $g$. Using the velocity $v(t)$ at the explicit time $t = 0.9905$ instead of the implicit $v(\tau-)$, again a different value for the sensitivity is obtained (a value again ignoring the changes in $\tau$ and not the one we are looking for), even though $\tau = 0.9905$ in both cases:&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L}{\text{d} g} = \begin{cases}
--0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\&lt;br&gt;
+-0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\
 -\frac{g}{\sqrt{v_0^2 + 2 g z_0}} = -0.99995 &amp;amp; \text{ with } v(t) = v_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -899,51 +796,48 @@ $$&lt;/p&gt;
 &lt;p&gt;We write&lt;/p&gt;
 &lt;p&gt;$$\text{solve}(t_0, x_0, t, p)$$&lt;/p&gt;
 &lt;p&gt;$(= x(t))$ for the functional solution of the ODE at time $t$.&lt;/p&gt;
-&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
+&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
 &lt;p&gt;Let us focus on the &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; algorithm which computes the sensitivities&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\&lt;br&gt;
+\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\
 \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}p} &amp;amp;= \lambda_{p}(t_{0}),
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;of a loss function $L$ acting on the final state with respect to the initial state and the parameters. It does so by solving an ODE for $\lambda(s)$ in reverse time from $t$ to $t_0$&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\&lt;br&gt;
+\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\
 \frac{\text{d}\lambda_{p}(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(x(s), \rightarrow p, s)}{\text{d} p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;with initial conditions:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\&lt;br&gt;
+\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\
 \lambda_{p}(t) &amp;amp;= 0.
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The arrows ($\rightarrow$) indicate the variable with respect to which we differentiate, which will become important later when the same variable shows up in multiple function arguments.&lt;/p&gt;
-&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in &lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;&lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, essentially the custom primitive differentiation rule of &lt;code&gt;solve&lt;/code&gt;, is the elementary building block needed to derive sensitivities also in more complicated examples:&lt;/p&gt;
 &lt;p&gt;Consider a loss depending on the state $x(s_i)$ at fix time points $s_i$ through loss functions $L_i^{\text{exp}}$,&lt;/p&gt;
 &lt;p&gt;$$
 L^{\text{exp}} = \sum_i L_i^{\text{exp}}(s_i, x(s_i), p).
 $$&lt;/p&gt;
 &lt;p&gt;Without contortions we can obtain the sensitivity of $L^{\text{exp}}$ in $p$ (or in $x_0$) using the tool we have. For those a bit familiar with automatic differentiation, this is perhaps easiest to see if we write $L^{\text{exp}}$ as pseudo code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1 = solve(t0, x0, s1, p)
-    L = L1(s1, x1, p)
-    x2 = solve(s1, x1, s2, p)
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\&lt;br&gt;
+\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\
 \frac{\text{d}\lambda_{p}(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(x(t), \rightarrow p, t)}{\text{d} p} - \frac{\text{d} L_i^{\text{exp}}(  x(t),\rightarrow p)}{\text{d} p}^\dagger \delta(t-s_i).
 \end{aligned}
 $$&lt;/p&gt;
@@ -957,24 +851,23 @@ $$&lt;/p&gt;
 $$&lt;/p&gt;
 &lt;p&gt;Now we can even properly define the &lt;code&gt;rrule&lt;/code&gt; of &lt;code&gt;solve&lt;/code&gt; in the sense of &lt;code&gt;DiffRules.jl&lt;/code&gt;.&lt;/p&gt;
 &lt;h3 id=&#34;explicit-events-1&#34;&gt;Explicit events&lt;/h3&gt;
-&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1- = solve(t0, x0, s1, p)
-    L = L1-(s1, x1-, p) # saved before affect
-    x1 = a(x1-, p, s1) # affect
-    L += L1(s1, x1, p) # saved after affect
-    x2- = solve(s1, x1, s2, p)
-    L += L2-(s2, x2-, p)
-    x2 = a(x2-, p, s2) # affect
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
+&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref2:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved before affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved after affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\&lt;br&gt;
+\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\
 \lambda_p({s_j}-) &amp;amp;= \lambda_p({s_j}) -  \lambda({s_j})^\dagger \frac{\text{d} a(x({s_j}-), \rightarrow p, {s_j}-)}{\text{d} p}
 \end{aligned}
 $$&lt;/p&gt;
@@ -988,27 +881,26 @@ $$&lt;/p&gt;
 &lt;p&gt;to put emphasis on this, or equivalently, compute for a unspecified function $L(t, x, p)$ the result of $\frac{\text{d}L}{\text{d}p}$ with&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\&lt;br&gt;
+\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\
 &amp;amp;= \frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;which indicates that changing $p$ influences $L$ both through changes in $\tau_1$ as well as changes in&lt;/p&gt;
 &lt;p&gt;$$\xi_1 = x(\tau_1-).$$&lt;/p&gt;
 &lt;p&gt;This case where we have a loss function $L = L_1$ depending on $\tau_1$, $x(\tau_1)$, and $p$
-was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;p&gt;Therefore, the sensitivity of the event time with respect to parameters $\frac{\text{d}\tau}{\text{d}p}$ must be taken into account.
-Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the 
-&lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
+Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the &lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\&lt;br&gt;
+\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The total derivative&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; inside the bracket is:
 $$
 \begin{aligned}
-\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\&lt;br&gt;
+\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;Since&lt;/p&gt;
@@ -1031,8 +923,8 @@ for the second term of $\dfrac{\text{d}\tau_1(p)}{\text{d}p}$.&lt;/p&gt;
 &lt;p&gt;We can now write the gradient as:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\&lt;br&gt;
-+&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\&lt;br&gt;
+\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\
++&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\
 +&amp;amp; \frac{\text{d}L(\rightarrow \tau_1(p), \text{solve}(t_0, x_0, \rightarrow \tau_1(p),  p), p)}{\text{d}\tau_1} \frac{\text{d} \tau_1(p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
@@ -1101,7 +993,7 @@ $$&lt;/p&gt;
 &lt;li&gt;If both limits contribute to the loss function, the contributions ${\color{red}\lambda_1}$ and ${\color{blue}\lambda_{a,1}}$ are added.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h4 id=&#34;generalization-several-events&#34;&gt;Generalization: several events&lt;/h4&gt;
-&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
+&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
 &lt;p&gt;$$
 \text{solve}(t_0, x_0, t, p)  = \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p),
 $$&lt;/p&gt;
@@ -1109,16 +1001,15 @@ $$&lt;/p&gt;
 &lt;p&gt;(A good exercise to get familiar with these type of arguments is to verify&lt;/p&gt;
 &lt;p&gt;$$\frac{\text{d}}{\text{d}s} \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p) = 0.)$$&lt;/p&gt;
 &lt;p&gt;Essentially, we know how to address several events already by considering a loss function&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss2(t0, x0, p)
-    tau1, xi1 = solve2(t0, x0, g, p)
-    L = L1(tau1, xi1, p)
-    tau2, xi2 = solve2(tau1, xi1, g, p)
-    L += L2(tau2, xi2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
 &lt;p&gt;Differentiating by hand, we have the following modification of the method from the previous section, which can be derived choosing a particular function $L$ in the previous section which incorporates the &lt;code&gt;solve&lt;/code&gt; from $\tau_1$ to $t_{\text{end}}$. Note&lt;/p&gt;
 &lt;p&gt;$$
 \lambda_{\tau_1}^0 = \frac{\text{d}(\text{solve}(\rightarrow \tau_1,  a(\xi_1,p), t_{\text{end}}, p))}{\text{d} \tau_1} = - f(a(\xi_1,p), p, \tau_1).
@@ -1150,43 +1041,38 @@ $$&lt;/p&gt;
 &lt;li&gt;Backpropagate $\lambda_J = {\color{red}\lambda_J} + {\color{blue}\lambda_{a,J}} + {\color{green}\lambda_{c,J}}$ to the next event time $\tau_{J-1}$ and iterate over the remaining $J-1$ events.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
-&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
-&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (
-&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$. &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$.&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the &lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
 $$
 \begin{aligned}
-\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\&lt;br&gt;
-&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\&lt;br&gt;
+\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\
+&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\
 &amp;amp;+ \frac{\text{d}f(t, x_1(t),x_2(\rightarrow t),\dots,x_N(t))}{\text{d}t} + \dots +  \frac{\text{d}f(t, x_1(t),x_2(t),\dots,x_N(\rightarrow t))}{\text{d}t}.
 \end{aligned}
-$$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+$$&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1196,8 +1082,7 @@ $$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-
       <pubDate>Fri, 02 Jul 2021 11:08:22 +0200</pubDate>
       <guid>https://frankschae.github.io/post/shadowing/</guid>
       <description>&lt;p&gt;In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &amp;ldquo;butterfly effect&amp;rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.&lt;/p&gt;
-&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
+&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:&lt;/p&gt;
 &lt;p&gt;$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -1211,57 +1096,54 @@ Under the assumption of ergodicity, $\langle g \rangle_∞$ only depends on $p$.
 &lt;p&gt;One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{pmatrix}
-\text{d}x \\&lt;br&gt;
-\text{d}y \\&lt;br&gt;
-\text{d}z \\&lt;br&gt;
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\&lt;br&gt;
-x(\rho-z) - y\\&lt;br&gt;
-x y - \beta z \\&lt;br&gt;
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
-
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
-
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &amp;quot;initial&amp;quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&amp;quot;attractor&amp;quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Statistics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;QuadGK&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SparseArrays&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# simulate 1 trajectory of the Lorenz system forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;28.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;initial&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;attractor&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
-  
 
 
 
@@ -1273,79 +1155,73 @@ savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; &gt;
 
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.&lt;/p&gt;
-&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
+&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
 &lt;p&gt;$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function compute_objective(sol)
-  quadgk(t-&amp;gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&amp;quot;z(t)&amp;quot;, xlabel=L&amp;quot;t&amp;quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&amp;quot;\langle z\rangle&amp;quot;, lw = 2)
-savefig(pl2, &amp;quot;zsingle.png&amp;quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
-
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
-
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&amp;quot;\langle z\rangle&amp;quot;, xlabel=L&amp;quot;\rho&amp;quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &amp;quot;zvsrho.png&amp;quot;)
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;quadgk&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;atol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;mean_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zsingle.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# for each value of the parameter, solve 20 times the initial value problem&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# wrap the procedure inside a function depending on p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ribbon&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;ho&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zvsrho.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;margin&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;600&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;300&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;z.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We obtain:&lt;/p&gt;
 
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &amp;quot;z.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We obtain:&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -1357,40 +1233,31 @@ savefig(pl4, &amp;quot;z.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/z.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/z.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;z.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$&lt;/p&gt;
 &lt;h2 id=&#34;conventional-forward-mode-sensitivity-analysis-and-finite-differencing&#34;&gt;Conventional forward-mode sensitivity analysis and finite-differencing&lt;/h2&gt;
-&lt;p&gt;For non-chaotic systems, we would just use the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we find diverging values:&lt;/p&gt;
+&lt;p&gt;For non-chaotic systems, we would just use the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_forward&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_calculus&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we find diverging values:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\&lt;br&gt;
+&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$&lt;/p&gt;
@@ -1398,45 +1265,37 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$&lt;/p&gt;
-&lt;p&gt;Similarly, using 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-     xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&amp;quot;Float32&amp;quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
-
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&amp;quot;Float32&amp;quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-
-savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
-
+&lt;p&gt;Similarly, using &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;500&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_Floats.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -1447,18 +1306,17 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Without animation:&lt;/p&gt;
 
@@ -1466,7 +1324,6 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-  
 
 
 
@@ -1478,60 +1335,50 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; &gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_Floats.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;Luckily, the 
-&lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
+&lt;p&gt;Luckily, the &lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
 &lt;blockquote&gt;
 &lt;p&gt;Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&amp;ldquo;shadows&amp;rdquo;) the numerically computed one.&lt;/p&gt;
 &lt;/blockquote&gt;
 &lt;h2 id=&#34;shadowing-methods&#34;&gt;Shadowing methods&lt;/h2&gt;
-&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref2:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref3:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;lss-and-nilss-for-the-lorenz-system&#34;&gt;LSS and NILSS for the Lorenz system&lt;/h2&gt;
-&lt;p&gt;Switching to LSS or NILSS within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
+&lt;p&gt;Switching to LSS or NILSS within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# objective&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# LSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;CosWindowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0095888187322035&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Cos2Windowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0343951385924328&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0284286902740765&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_adjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.028428690274077&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# select via sensealg in solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9694728321500617&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;&lt;code&gt;CosWindowing()&lt;/code&gt; (default)&lt;/li&gt;
 &lt;li&gt;&lt;code&gt;Cos2Windowing()&lt;/code&gt;&lt;/li&gt;
@@ -1540,72 +1387,68 @@ dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
 &lt;p&gt;Additionally, an adjoint implementation &lt;code&gt;AdjointLSS()&lt;/code&gt; is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that &lt;code&gt;AdjointLSS(alpha=10.0)&lt;/code&gt; agrees perfectly with &lt;code&gt;ForwardLSS(alpha=10.0)&lt;/code&gt;. In all cases considered, we find the expected sensitivity value of $\approx 1$.&lt;/p&gt;
 &lt;p&gt;However, the use of &lt;code&gt;LSS()&lt;/code&gt; is (typically) much more expensive than the use of &lt;code&gt;NILSS()&lt;/code&gt;, because &lt;code&gt;LSS()&lt;/code&gt; needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of &lt;code&gt;NILSS()&lt;/code&gt; scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making &lt;code&gt;NILSS()&lt;/code&gt; more efficient.&lt;/p&gt;
 &lt;p&gt;In the &lt;code&gt;NILSS()&lt;/code&gt; algorithm, the user can control the number of steps per segment as well as the number of segments.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# NILSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# make sure trajectory is fully on the attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;120.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9966924374966089&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0416028730638789&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Warning: Detected a large value of ξ at the beginning of a segment.&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref4:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;NILSAS&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;li&gt;FD-NILSS&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref4:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019) &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019)&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019) &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019)&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1614,21 +1457,11 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
       <link>https://frankschae.github.io/post/hybridde/</link>
       <pubDate>Wed, 16 Jun 2021 14:50:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/hybridde/</guid>
-      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
+      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
 &lt;p&gt;This first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).&lt;/p&gt;
-&lt;p&gt;Note: Please check the 
-&lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
+&lt;p&gt;Note: Please check the &lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
 &lt;h2 id=&#34;hybrid-differential-equations&#34;&gt;Hybrid Differential Equations&lt;/h2&gt;
-&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
+&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
 &lt;p&gt;Some relevant examples for hybrid DEs with discrete or continuous callbacks are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; (&lt;code&gt;ContinuousCallback&lt;/code&gt;).&lt;/li&gt;
@@ -1638,7 +1471,7 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
 &lt;li&gt;pharmacokinetic models&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, where explicit dosing times change the drug concentration in the blood (&lt;code&gt;DiscreteCallback&lt;/code&gt;). The simplest possible example being the one-compartment model.&lt;/li&gt;
 &lt;li&gt;kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (&lt;code&gt;DiscreteCallback&lt;/code&gt;).&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;kicked-harmonic-oscillator&#34;&gt;Kicked harmonic oscillator&lt;/h2&gt;
 &lt;p&gt;Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form&lt;/p&gt;
 &lt;p&gt;$$
@@ -1647,40 +1480,38 @@ $$&lt;/p&gt;
 &lt;p&gt;where $a=0.1$ and $b=1$ with initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-x(t=0) &amp;amp;= 1  \\&lt;br&gt;
+x(t=0) &amp;amp;= 1  \\
 v(t=0) &amp;amp;= \dot{x}(t=0) = 0.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;This second-order ODE can be 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use 
-&lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity
-using Zygote
-using Random
-u0 = Float32[1.; 0.]
-
-tspan = (0.0f0,50.0f0)
-
-dtsave = 0.5f0
-t = tspan[1]:dtsave:tspan[2]
-
-function oscillator!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  return nothing
-end
-
-prob_data = ODEProblem(oscillator!,u0,tspan)
+&lt;p&gt;This second-order ODE can be &lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use &lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqFlux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Optim&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# ODE without kicks&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-# ODE without kicks
-pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -1692,44 +1523,36 @@ pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &am
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as &lt;code&gt;1&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;kicktimes = tspan[1]:1:tspan[2]
-function kick!(integrator)
-  integrator.u[end] += one(eltype(integrator.u))
-end
-cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false))
-
-sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t)
-t_data = sol_data.t
-ode_data = Array(sol_data)
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualize data&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
-# visualize data
-pl1 = plot(t_data,ode_data[1,:],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl1,t_data,ode_data[2,:],label=&amp;quot;data v(t)&amp;quot;)
 
-pl2 = plot(t_data[1:20],ode_data[1,1:20],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl2,t_data[1:20],ode_data[2,1:20],label=&amp;quot;data v(t)&amp;quot;)
-pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;
 
 
 
 
-  
 
 
 
@@ -1741,63 +1564,48 @@ pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting &lt;code&gt;save_positions=(true,true)&lt;/code&gt;, the kicks would be saved before &lt;strong&gt;and&lt;/strong&gt; after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;, we now aim at learning (potentially) missing parts of the model from these data traces.&lt;/p&gt;
 &lt;h3 id=&#34;high-domain-knowledge&#34;&gt;High domain knowledge&lt;/h3&gt;
 &lt;p&gt;For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the &lt;code&gt;affect!&lt;/code&gt; function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of &lt;code&gt;integrator.u&lt;/code&gt;. If we further set the initial value of that component to &lt;code&gt;1&lt;/code&gt;, then the neural network only has to learn that &lt;code&gt;du[3]&lt;/code&gt; is &lt;code&gt;0&lt;/code&gt;. In other words, the output of the neural network must be &lt;code&gt;0&lt;/code&gt; for all states &lt;code&gt;u&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;Random.seed!(123)
-nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1))
-p_nn1 = initial_params(nn1)
-
-function f1!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3] = nn1(u[1:2], p)[1]
-  return nothing
-end
-
-affect!(integrator) = integrator.u[2] += integrator.u[3]
-cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false))
-z0 = Float32[u0;one(u0[1])]
-prob1 = ODEProblem(f1!,z0,tspan,p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# to visualize the predictions of the trained neural network below
-function visualize(prob,p)
-  _prob = remake(prob,p=p)
-  ode_pred = Array(solve(_prob,Tsit5(),callback=cb,
-                 saveat=dtsave))[1:2,:]
-  pl1 = plot(t_data,ode_pred[1,:],label=&amp;quot;x(t)&amp;quot;)
-  scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=&amp;quot;data x(t)&amp;quot;)
-  pl2 = plot(t_data,ode_pred[2,:],label=&amp;quot;v(t)&amp;quot;)
-  scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=&amp;quot;data v(t)&amp;quot;)
-
-  pl = plot(pl1, pl2, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-  return pl, sum(abs2,ode_data .- ode_pred)
-end
-
-pl = plot(solve(prob1,Tsit5(),saveat=t,
-  callback=cb
-  ),label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;123&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FastChain&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tanh&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial_params&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# to visualize the predictions of the trained neural network below&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                 &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -1809,76 +1617,65 @@ pl = plot(solve(prob1,Tsit5(),saveat=t,
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;untrained_nn.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;which (of course) doesn&amp;rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### loss function
-function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint())
-  _prob = remake(prob,p=p)
-  pred = Array(solve(_prob,Tsit5(),callback=cb,
-               saveat=dtsave,sensealg=sensealg))[1:2,:]
-  sum(abs2,ode_data .- pred)
-end
-
-loss(p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The recently implemented tools are deeply hidden within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., 
-&lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;###################################
-# training loop
-# optimize the parameters for a few epochs with ADAM
-function train(prob, p_nn; sensealg=BacksolveAdjoint())
-  opt = ADAM(0.0003f0)
-  list_plots = []
-  losses = []
-  for epoch in 1:200
-    println(&amp;quot;epoch: $epoch / 200&amp;quot;)
-    _dy, back = Zygote.pullback(p -&amp;gt; loss(p,
-      prob=prob,
-      sensealg=sensealg), p_nn)
-    gs = @time back(one(_dy))[1]
-    push!(losses, _dy)
-    if epoch % 10 == 0
-      # plot every xth epoch
-      pl, test_loss = visualize(prob, p_nn)
-      println(&amp;quot;Loss (epoch: $epoch): $test_loss&amp;quot;)
-      display(pl)
-      push!(list_plots, pl)
-    end
-    Flux.Optimise.update!(opt, p_nn, gs)
-    println(&amp;quot;&amp;quot;)
-  end
-  return losses, list_plots
-end
-
-# plot training loss
-losses, list_plots = train(prob1, p_nn1)
-pl1 = plot(losses, lw = 1.5, xlabel = &amp;quot;epoch&amp;quot;, ylabel=&amp;quot;loss&amp;quot;, legend=false)
-pl2 = list_plots[end]
-pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t,
-   callback=cb
-  ), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-
-pl = plot(pl2,pl3)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### loss function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ReverseDiffAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;               &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The recently implemented tools are deeply hidden within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., &lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;###################################&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# training loop&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# optimize the parameters for a few epochs with ADAM&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ADAM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0003f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;200&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt; / 200&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;%&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;c&#34;&gt;# plot every xth epoch&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_loss&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Loss (epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;): &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$test_loss&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;display&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Optimise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;update!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot training loss&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;epoch&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;loss&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -1890,98 +1687,90 @@ pl = plot(pl2,pl3)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/trained1.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/trained1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;trained1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We see the expected constant value of &lt;code&gt;u[3]&lt;/code&gt;, indicating a kick to the velocity of &lt;code&gt;+=1&lt;/code&gt;, at the kicking times over the full time interval.&lt;/p&gt;
 &lt;h2 id=&#34;reducing-the-domain-knowledge&#34;&gt;Reducing the domain knowledge&lt;/h2&gt;
-&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
+&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;changing the initial condition of the third component of &lt;code&gt;u&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;using another affect function &lt;code&gt;affect!(integrator) = integrator.u[2] = integrator.u[3]&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;dropping the knowledge that only &lt;code&gt;u[2]&lt;/code&gt; gets a kick by using a neural network with &lt;code&gt;2&lt;/code&gt; outputs (+ a fourth component in the ODE):&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4]
-function f2!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;fitting the parameters $a$ and $b$ simultaneously:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f3!(du,u,p,t)
-  a = p[end-1]
-  b = p[end]
-  nn_weights = p[1:end-2]
-
-  du[1] = u[2]
-  du[2] = -b*u[1] - a*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f3!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;inferring the entire underlying dynamics using a neural network with &lt;code&gt;4&lt;/code&gt; outputs:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f4!(du,u,p,t)
-  Ω = nn3(u[1:2], p)
-
-  du[1] = Ω[1]
-  du[2] = Ω[2]
-  du[3:4] .= Ω[3:4]
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f4!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;etc.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;refine the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;) and&lt;/li&gt;
-&lt;li&gt;support direct usage through the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
+&lt;li&gt;support direct usage through the &lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021) &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021)&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1991,152 +1780,100 @@ end
       <pubDate>Wed, 26 Aug 2020 22:59:49 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc-2020/</guid>
       <description>&lt;h2 id=&#34;project-summary&#34;&gt;Project summary&lt;/h2&gt;
-&lt;p&gt;In this project, we have implemented new promising tools within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
+&lt;p&gt;In this project, we have implemented new promising tools within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
 The high weak order solvers will allow for massive performance advantages for fitting expectations of equations.
 Instead of automatic differentiation (AD) through the operations of an SDE solver, which scales poorly in memory, one can now use efficient stochastic adjoint sensitivity methods.&lt;/p&gt;
 &lt;h2 id=&#34;blog-posts&#34;&gt;Blog posts&lt;/h2&gt;
 &lt;p&gt;The following posts describe the work during the entire period in more detail:&lt;/p&gt;
 &lt;ol&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GSoC 2020: High weak order SDE solvers and their utility in neural SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/high-weak/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;High weak order SDE solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GSoC 2020: High weak order SDE solvers and their utility in neural SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/high-weak/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;High weak order SDE solvers&lt;/a&gt;&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;docs&#34;&gt;Docs&lt;/h2&gt;
-&lt;p&gt;The documentation of the solvers is available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/solvers/sde_solve/#High-Weak-Order-Methods&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.
-Docs with respect to the adjoint sensitivity tools will be available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The documentation of the solvers is available &lt;a href=&#34;https://diffeq.sciml.ai/latest/solvers/sde_solve/#High-Weak-Order-Methods&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.
+Docs with respect to the adjoint sensitivity tools will be available &lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;achievements&#34;&gt;Achievements&lt;/h2&gt;
 &lt;p&gt;Please find below a list of the PRs carried out during GSoC in the different repositories in chronological order within the SciML ecosystem.&lt;/p&gt;
 &lt;h4 id=&#34;stochasticdiffeqjl&#34;&gt;StochasticDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/285&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Inplace version of DRI1 scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/289&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI1 method&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;tstop fixes for reverse time propagation&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes for EulerHeun scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/327&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Speed up the tests for the DRI1 and RI1 schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/328&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/329&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RDI1WM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/332&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adaptive version of the stochastic Runge-Kutta schemes by embedding&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/333&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RS1 and RS2 schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/334&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PL1WM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/337&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NON scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/338&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for weak methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/342&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stochastic improved and modified Euler methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/343&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;COM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/348&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Computationally more efficient NON variant (NON2)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/285&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Inplace version of DRI1 scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/289&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI1 method&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;tstop fixes for reverse time propagation&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes for EulerHeun scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/327&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Speed up the tests for the DRI1 and RI1 schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/328&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/329&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RDI1WM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/332&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adaptive version of the stochastic Runge-Kutta schemes by embedding&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/333&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RS1 and RS2 schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/334&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PL1WM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/337&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NON scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/338&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for weak methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/342&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stochastic improved and modified Euler methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/343&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;COM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/348&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Computationally more efficient NON variant (NON2)&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/288&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Static array tests&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Levy area for non-commutative noise processes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/288&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Static array tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Levy area for non-commutative noise processes&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqsensitivityjl&#34;&gt;DiffEqSensitivity.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/235&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint sensitivities for steady states&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/237&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Concrete_solve dispatch for steady state problem&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/242&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;BacksolveAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/256&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GPU savety for SDE BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/258&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Tests for concrete solve with respect to SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/260&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Alternative differentiation choices (vjps) for noise Jacobian&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/265&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes and tests for inplace formulation of BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/268&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Efficient SDE BacksolveAdjoint for scalar noise&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/275&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/295&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;InterpolatingAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/298&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for backsolve, steadystate and interpolation adjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/299&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for more general noise processes: replace NoiseGrid by NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/303&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Cheaper non-diagonal noise tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/235&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint sensitivities for steady states&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/237&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Concrete_solve dispatch for steady state problem&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/242&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;BacksolveAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/256&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GPU savety for SDE BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/258&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Tests for concrete solve with respect to SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/260&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Alternative differentiation choices (vjps) for noise Jacobian&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/265&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes and tests for inplace formulation of BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/268&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Efficient SDE BacksolveAdjoint for scalar noise&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/275&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/295&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;InterpolatingAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/298&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for backsolve, steadystate and interpolation adjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/299&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for more general noise processes: replace NoiseGrid by NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/303&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Cheaper non-diagonal noise tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support adjoints for SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support adjoints for SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqnoiseprocessjl&#34;&gt;DiffEqNoiseProcess.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/48&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Multi-dimensional Brownian motion tests&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/49&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Bug fix for inplace form of NoiseGrid&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/51&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversible NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/53&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Relax the size constraints of the available noise processes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/54&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the real-valued white noise process function&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/55&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix of an extraction issue with NoiseGrid&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Extraction and endpoint fixes on NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversal of SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/48&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Multi-dimensional Brownian motion tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/49&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Bug fix for inplace form of NoiseGrid&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/51&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversible NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/53&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Relax the size constraints of the available noise processes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/54&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the real-valued white noise process function&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/55&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix of an extraction issue with NoiseGrid&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Extraction and endpoint fixes on NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversal of SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqgpujl&#34;&gt;DiffEqGPU.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl/pull/59&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Memory efficient reduction function for ensemble problems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl/pull/59&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Memory efficient reduction function for ensemble problems&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;modelingtoolkitjl&#34;&gt;ModelingToolkit.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqdevtoolsjl&#34;&gt;DiffEqDevTools.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NoiseWrapper alternative for analyticless convergence tests of SDE solvers&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/74&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;test_convergence() dispatch for ensemble simulations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Work precision set for ensemble problems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NoiseWrapper alternative for analyticless convergence tests of SDE solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/74&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;test_convergence() dispatch for ensemble simulations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Work precision set for ensemble problems&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqbasejl&#34;&gt;DiffEqBase.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqBase.jl/pull/503&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix concrete_solve tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqBase.jl/pull/503&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix concrete_solve tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;future-work&#34;&gt;Future work&lt;/h2&gt;
 &lt;p&gt;There is still a lot that we&amp;rsquo;d like to do, e.g.,&lt;/p&gt;
@@ -2150,12 +1887,8 @@ Docs with respect to the adjoint sensitivity tools will be available
 &lt;/ul&gt;
 &lt;p&gt;Contributions, suggestions &amp;amp; comments are always welcome! You might like to join our slac channels #diffeq-bridged and #neuralsde to get in touch.&lt;/p&gt;
 &lt;h2 id=&#34;acknowledgement&#34;&gt;Acknowledgement&lt;/h2&gt;
-&lt;p&gt;I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback.
-I would also like to thank 
-&lt;a href=&#34;https://www.quantumtheory-bruder.physik.unibas.ch/people.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts.
+&lt;p&gt;I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback.
+I would also like to thank &lt;a href=&#34;https://www.quantumtheory-bruder.physik.unibas.ch/people.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts.
 Many thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
 </description>
     </item>
@@ -2165,11 +1898,8 @@ Many thanks to the very supportive julia community and to Google&amp;rsquo;s ope
       <link>https://frankschae.github.io/post/high-weak/</link>
       <pubDate>Mon, 17 Aug 2020 14:46:35 +0200</pubDate>
       <guid>https://frankschae.github.io/post/high-weak/</guid>
-      <description>&lt;p&gt;This post summarizes our new 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
-for the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
+      <description>&lt;p&gt;This post summarizes our new &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
+for the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
 &lt;p&gt;After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
 Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics.
 Finally, we discuss how to run simulations on GPU hardware.&lt;/p&gt;
@@ -2200,16 +1930,14 @@ $$
 \rm{E}(|X(T) -Y(T)|) \leq C \cdot h^p,
 $$
 for each $h \in [0, \delta]$.&lt;/p&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
+&lt;p&gt;The &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
 for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms.
 For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI)
 can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where &lt;em&gt;b&lt;/em&gt; has only entries on its diagonal or $m=1$.
 The main difficulty for the construction of strong methods with an order &amp;gt; 1/2 arises from the need of an accurate estimation of
 multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of &lt;em&gt;dW&lt;/em&gt; in the case
 of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed
-in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also 
-&lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
+in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also &lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;weak-convergence&#34;&gt;Weak convergence&lt;/h2&gt;
 &lt;p&gt;Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt; in many situations. In these cases, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient and &amp;ndash; due to the less restrictive formulation of the objective &amp;ndash; those solvers are computationally cheaper than their strong counterparts.
 For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics.
@@ -2224,19 +1952,17 @@ $~$&lt;/p&gt;
 &lt;p&gt;for any polynomial $g$ for each $h \in [0, \delta]$.&lt;/p&gt;
 &lt;p&gt;We demonstrate below that &lt;strong&gt;high weak order solvers&lt;/strong&gt; are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.&lt;/p&gt;
 &lt;h2 id=&#34;new-high-weak-order-methods&#34;&gt;New high weak order methods&lt;/h2&gt;
-&lt;p&gt;A list of all new weak solvers is available in the 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
+&lt;p&gt;A list of all new weak solvers is available in the &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
 Note that we also implemented methods designed for the Stratonovich sense.
 For the subsequent examples regarding Ito SDEs, we use only a subset of the plethora of second-order weak solvers.
-We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
-We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
+We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
+We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref2:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref3:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
 The latter is the simplest weak solver, where the Gaussian increments of the strong Euler-Maruyama scheme are replaced by
 two-point distributed random variables with similar moment properties.&lt;/p&gt;
 &lt;p&gt;Rößler&amp;rsquo;s SRK schemes are particularly designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;, since only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn and since the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.
 &lt;code&gt;PL1WM()&lt;/code&gt; in contrast needs to simulate &lt;code&gt;m(m+1)/2&lt;/code&gt; random variables but a smaller number of order conditions needs to be fulfilled.&lt;/p&gt;
 &lt;h3 id=&#34;convergence-tests&#34;&gt;Convergence tests&lt;/h3&gt;
-&lt;p&gt;As in the 
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As in the &lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -2247,67 +1973,61 @@ $$ ~$$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;for the function $f(x)=(x_1)^2$, which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e6&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-using Test
-using Random
-using Plots
-using DiffEqDevTools
-
-function prob_func(prob, i, repeat)
-    remake(prob,seed=seeds[i])
-end
-
-u₀ = [1.0,1.0]
-function f1!(du,u,p,t)
-  @inbounds begin
-    du[1] = -273//512*u[1]
-    du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-  end
-  return nothing
-end
-function g1!(du,u,p,t)
-  @inbounds begin
-    du[1,1] = 1//4*u[1]
-    du[1,2] = 1//16*u[1]
-    du[2,1] = (1-2*sqrt(2))/4*u[1]
-    du[2,2] = 1//10*u[1]+1//16*u[2]
-  end
-  return nothing
-end
-dts = 1 .//2 .^(3: -1:0)
-tspan = (0.0,3.0)
-
-h2(z) = z^2 # but apply it only to u[1]
-
-prob = SDEProblem(f1!,g1!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-numtraj = Int(1e6)
-seed = 100
-Random.seed!(seed)
-seeds = rand(UInt, numtraj)
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h2(sol[end][1]),false),
-        prob_func = prob_func
-        )
-
-sim = test_convergence(dts,ensemble_prob,DRI1(),
-  save_everystep=false,trajectories=numtraj,save_start=false,adaptive=false,
-  weak_timeseries_errors=false,weak_dense_errors=false,
-  expected_value=exp(-3.0)
-  )
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqDevTools&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# but apply it only to u[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Int&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_convergence&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
 Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function for the other aforementioned solvers, we obtain the convergence plot:&lt;/p&gt;
 
 
 
 
 
-  
-
-
-
-
 
 
 
@@ -2315,18 +2035,16 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/weak_conv.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/weak_conv.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;weak_conv.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Note that the &lt;code&gt;SimplifiedEM&lt;/code&gt; and the &lt;code&gt;EM&lt;/code&gt; scheme fall on top of each other.
 &lt;code&gt;DRI1()&lt;/code&gt; achieves the smallest errors for a fixed &lt;code&gt;dt&lt;/code&gt; in this study.&lt;/p&gt;
@@ -2334,35 +2052,29 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 &lt;p&gt;Ultimately, we are not only interested in the general convergence slope of an algorithm but also in its speed.
 We&amp;rsquo;d like to select the fastest method depending on the permitted tolerance.
 This is commonly studied using work-precision diagram.
-Thanks to 
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;reltols = 1.0 ./ 4.0 .^ (1:4)
-abstols = reltols#[0.0 for i in eachindex(reltols)]
-setups = [
-          Dict(:alg=&amp;gt;DRI1(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;PL1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;EM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;SimplifiedEM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI2WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false)
-          ]
-test_dt = 1//10000
-appxsol_setup = Dict(:alg=&amp;gt;EM(), :dt=&amp;gt;test_dt)
-wp = @time WorkPrecisionSet(ensemble_prob,
-                            abstols,reltols,setups,test_dt;
-                            maxiters = 1e7,verbose=false,
-                            save_everystep=false,save_start=false,
-                            appxsol_setup = appxsol_setup,
-                            expected_value=exp(-3.0),
-                            trajectories=numtraj, error_estimate=:weak_final)
-plt = plot(wp;legend=:bottomleft)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
+Thanks to &lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;./&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;4.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;c&#34;&gt;#[0.0 for i in eachindex(reltols)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;PL1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;SimplifiedEM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI2WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10000&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WorkPrecisionSet&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;maxiters&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;verbose&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;error_estimate&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:weak_final&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-  
 
 
 
@@ -2374,88 +2086,84 @@ plt = plot(wp;legend=:bottomleft)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;WorkPrecision.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Therefore, &lt;code&gt;DRI1&lt;/code&gt; has the best performance in this non-commutative noise case if the error is supposed to stay below 1e-3.
 For larger permitted errors, the &lt;code&gt;SimplifiedEM&lt;/code&gt; scheme might be a good choice. However, the first order methods
 are outclassed when high precision is more of a concern.
-We plan to perform more in-depth benchmarks in the near future what will be reported on the 
-&lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
+We plan to perform more in-depth benchmarks in the near future what will be reported on the &lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
 &lt;h3 id=&#34;adaptive-step-size-control&#34;&gt;Adaptive step-size control&lt;/h3&gt;
-&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
+&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
 &lt;p&gt;To use adaptive step-size control, it is sufficient to set &lt;code&gt;adaptive=true&lt;/code&gt; (default setting). Optionally, one may also pass absolute and relative tolerances.&lt;/p&gt;
 &lt;p&gt;The following julia code simulates the stochastic version of the Brusselator equations with intitial condition&lt;/p&gt;
 &lt;p&gt;$$ X(t=0)=  \begin{pmatrix} 0.1 \\ 0\end{pmatrix},$$&lt;/p&gt;
 &lt;p&gt;on a time span $\mathbb{I}=[0, 100]$ for adaptive (&lt;code&gt;sol&lt;/code&gt;) and fixed step sizes (&lt;code&gt;sol_na&lt;/code&gt;):&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq, DiffEqNoiseProcess, Random
-using Plots
-using DiffEqGPU
-
-function prob_func(prob, i, repeat)
-  Random.seed!(seeds[i])
-  W = WienerProcess(0.0,0.0,0.0)
-  remake(prob,noise=W)
-end
-
-function brusselator_f!(du,u,p,t)
-  @inbounds begin
-    du[1] = (p[1]-1)*u[1]+p[1]*u[1]^2+(u[1]+1)^2*u[2]
-    du[2] = -p[1]*u[1]-p[1]*u[1]^2-(u[1]+1)^2*u[2]
-  end
-  nothing
-end
-
-function scalar_noise!(du,u,p,t)
-  @inbounds begin
-    du[1] = p[2]*u[1]*(1+u[1])
-    du[2] = -p[2]*u[1]*(1+u[1])
-   end
-   nothing
-end
-
-
-# fix seeds
-seed = 100
-Random.seed!(seed)
-numtraj= 100
-seeds = rand(UInt, numtraj)
-W = WienerProcess(0.0,0.0,0.0)
-
-
-u0 = [-0.1f0,0.0f0]
-tspan = (0.0f0,100.0f0)
-p = [1.9f0,0.1f0]
-
-prob = SDEProblem(brusselator_f!,scalar_noise!,u0,tspan,p,noise=W)
-
-ensembleprob = EnsembleProblem(prob, prob_func = prob_func)
-
-sol = @time solve(ensembleprob,DRI1(),dt=0.1,EnsembleCPUArray(),trajectories=numtraj)
-sol_na = @time solve(ensembleprob,DRI1(),dt=0.8,adaptive=false,EnsembleCPUArray(),trajectories=numtraj)
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqGPU&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# fix seeds&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.9f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleSummary&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fillalpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;time t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;X(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
 
-summ = EnsembleSummary(sol,0.0f0:0.5f0:100f0)
-pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quot;X(t)&amp;quot;, label= [&amp;quot;x₁(t)&amp;quot; &amp;quot;x₂(t)&amp;quot;], legend=true)
 
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -2466,69 +2174,53 @@ pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quo
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator_many_trajectories.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
+&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref3:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
 as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DifferentialEquations.EnsembleAnalysis
-meansol = timeseries_steps_mean(sol)
-meansol_na = timeseries_point_mean(sol_na,meansol.t)
-dts = []
-tmp1 = tspan[1]
-for tmp2 in meansol.t
-  global tmp1
-  push!(dts,tmp2-tmp1)
-  tmp1 = tmp2
-end
-#
-
-list_plots = []
-for i in 1:length(meansol.u)
-  l = @layout [a  b]
-  plt1 = plot(meansol[1, 1:i],meansol[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;adaptive&amp;quot;,
-         lw=2,
-         linecolor=1
-         )
-  plot!(meansol_na[1, 1:i],meansol_na[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;fixed step size&amp;quot;,
-         lw=2,
-         linecolor=2
-         )
-  pl2 = scatter(dts[1:i], xlabel = &amp;quot;step&amp;quot;, yaxis= &amp;quot;dtᵢ&amp;quot;, xlim = (0, length(meansol.u)),  ylim = (0.0, 2.3), legend=false)
-  plt = plot(plt1, pl2, layout = l)
-  push!(list_plots, plt)
-end
-
-anim = animate(list_plots,lw=2,every=1)
-
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleAnalysis&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_steps_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_point_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;global&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;#&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@layout&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;adaptive&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;fixed step size&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scatter&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;step&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;dtᵢ&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)),&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;2.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -2540,53 +2232,48 @@ anim = animate(list_plots,lw=2,every=1)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gpu-usage&#34;&gt;GPU usage&lt;/h3&gt;
 &lt;p&gt;All necessary tools to accelerate the simulation of (stochastic) differential equations on GPUs within the SciML
-ecosystem are collected in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
+ecosystem are collected in the &lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;Currently, bounds checking and return values are not allowed, i.e., functions must be
 written in the form:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f!(du,u,p,t)
-  @inbounds begin
-    du[1] = ..
-  end
-  nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;..&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
 Note that for some high weak order solvers GPU usage is not recommended as scalar indexing is used.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -2595,21 +2282,14 @@ Note that for some high weak order solvers GPU usage is not recommended as scala
       <link>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</link>
       <pubDate>Sat, 30 May 2020 15:10:33 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</guid>
-      <description>&lt;p&gt;First and foremost, I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
+      <description>&lt;p&gt;First and foremost, I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
 Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both  &lt;strong&gt;new high weak order solvers&lt;/strong&gt; as well as &lt;strong&gt;adjoint sensitivity methods&lt;/strong&gt; for stochastic differential equations (SDEs).
-The project is embedded within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
-Ultimately, the planned contributions will allow researchers to simulate (or even 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
+The project is embedded within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
+Ultimately, the planned contributions will allow researchers to simulate (or even &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
 Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.&lt;/p&gt;
 &lt;p&gt;This post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.&lt;/p&gt;
 &lt;h2 id=&#34;high-weak-order-solvers&#34;&gt;High Weak Order Solvers&lt;/h2&gt;
-&lt;p&gt;Currently, the 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
+&lt;p&gt;Currently, the &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
 In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.&lt;/p&gt;
 &lt;p&gt;However in many situations, we are only aiming for computing an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt;.
 In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods.
@@ -2617,7 +2297,7 @@ In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&
 As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.&lt;/p&gt;
 &lt;h3 id=&#34;second-order-runge-kutta-methods-for-ito-sdes&#34;&gt;Second order Runge-Kutta methods for Ito SDEs&lt;/h3&gt;
 &lt;p&gt;In the beginning of the community bonding period I finished the implementations of the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and &lt;code&gt;RI1()&lt;/code&gt;&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;. Specifically, only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn (in contrast to &lt;code&gt;m(m+1)/2&lt;/code&gt; from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.&lt;/p&gt;
-&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -2626,47 +2306,38 @@ $$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e7&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;. The code for a single  &lt;code&gt;dt&lt;/code&gt; reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-numtraj = 1e7
-u₀ = [1.0,1.0]
-function f!(du,u,p,t)
-  du[1] = -273//512*u[1]
-  du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-end
-function g!(du,u,p,t)
-  du[1,1] = 1//4*u[1]
-  du[1,2] = 1//16*u[1]
-  du[2,1] = (1-2*sqrt(2))/4*u[1]
-  du[2,2] = 1//10*u[1]+1//16*u[2]
-end
-dt = 1//8
-tspan = (0.0,10.0)
-prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-h(z) = z^2
-
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h(sol[end][1]),false)
-        )
-sol = solve(ensemble_prob, DRI1();
-        dt=dt,
-        save_start=false,
-        save_everystep=false,
-        weak_timeseries_errors=false,
-        weak_dense_errors=false,
-        trajectories=numtraj)
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
-&lt;pre&gt;&lt;code&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
-
-
-
-
-
-  
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;();&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
 
 
 
@@ -2678,29 +2349,26 @@ sol = solve(ensemble_prob, DRI1();
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/DRI1.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/DRI1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;DRI1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In the next couple of weeks, my focus will be on&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;adding other high weak order solvers,&lt;/li&gt;
 &lt;li&gt;implementing adaptive time stepping.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;More of our near-term goals are collected in this 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;More of our near-term goals are collected in this &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;adjoint-sensitivity-methods-for-sdes&#34;&gt;Adjoint Sensitivity Methods for SDEs&lt;/h2&gt;
-&lt;p&gt;In 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
+&lt;p&gt;In &lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
 &lt;p&gt;$$
 L = |f(p)-y|^2.
 $$&lt;/p&gt;
@@ -2709,46 +2377,42 @@ One possibility to obtain the gradient information for (stochastic) differential
 While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.&lt;/p&gt;
 &lt;p&gt;Alternatively to the &amp;ldquo;direct&amp;rdquo; AD approaches, the &lt;strong&gt;adjoint sensitivity method&lt;/strong&gt; can be used&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.
 Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).&lt;/p&gt;
-&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
+&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
 &lt;h3 id=&#34;reverse-sde-time-evolution&#34;&gt;Reverse SDE time evolution&lt;/h3&gt;
 &lt;p&gt;The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms.
 As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed.
 In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values.
-After some fixes for the 
-&lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
+After some fixes for the &lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
 &lt;p&gt;As an example, the code below computes first the forward evolution of an SDE&lt;/p&gt;
 &lt;p&gt;$$ dX  =  \alpha X dt + \beta X dW$$&lt;/p&gt;
 &lt;p&gt;with $\alpha=1.01$, $\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;  using StochasticDiffEq, DiffEqNoiseProcess
-
-  α=1.01
-  β=0.87
-
-  dt = 1e-3
-  tspan = (0.0,1.0)
-  u₀=1/2
-
-  tarray =  collect(tspan[1]:dt:tspan[2])
-
-  f!(du,u,p,t) = du .= α*u
-  g!(du,u,p,t) = du .= β*u
-
-
-  prob = SDEProblem(f!,g!,[u₀],tspan)
-  sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false)
-
-  _sol = deepcopy(sol) # to make sure the plot is correct
-  W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W))
-  prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1)
-  sol1 = solve(prob1,EulerHeun(),dt=dt)
-&lt;/code&gt;&lt;/pre&gt;
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# to make sure the plot is correct&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NoiseGrid&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
-  
 
 
 
@@ -2760,18 +2424,12 @@ After some fixes for the
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/animation.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/animation.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;animation.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gradients-of-diagonal-sdes&#34;&gt;Gradients of diagonal SDEs&lt;/h3&gt;
 &lt;p&gt;I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.&lt;/p&gt;
@@ -2785,82 +2443,80 @@ $$&lt;/p&gt;
 &lt;p&gt;acting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\&lt;br&gt;
+\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\
 \frac{d L}{d \beta}  &amp;amp;= 2 \sum_i W(t_i) |X(t_i)|^2
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Test, LinearAlgebra
-using DiffEqSensitivity, StochasticDiffEq
-using Random
-
-seed = 100
-Random.seed!(seed)
-
-u₀ = [0.5]
-tstart = 0.0
-tend = 0.1
-dt = 0.005
-trange = (tstart, tend)
-t = tstart:dt:tend
-tarray = collect(t)
-
-function g(u,p,t)
-  sum(u.^2.0)
-end
-
-function dg!(out,u,p,t,i)
-  (out.=-2.0*u)
-end
-
-p2 = [1.01,0.87]
-
-f(u,p,t) = p[1]*u
-σ(u,p,t) = p[2]*u
-
-
-Random.seed!(seed)
-prob = SDEProblem(f,σ,u₀,trange,p2)
-sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true)
-res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint())
-
-
-noise = vec((@. sol.W(tarray)))
-Wextracted = [W[1][1] for W in noise]
-resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp = [resp1, resp2]
-
-@test isapprox(res_p&#39;, resp, rtol = 1e-6)
-# True
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.005&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:tend&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.=-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RKMil&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;interpretation&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:Stratonovich&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res_u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adjoint_sensitivities&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vec&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;isapprox&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# True&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;finish the current backsolve adjoint version,&lt;/li&gt;
 &lt;li&gt;allow for computing the gradients of non-commuting SDEs,&lt;/li&gt;
 &lt;li&gt;implement also an interpolation adjoint version,&lt;/li&gt;
 &lt;li&gt;benchmark it with respect to AD approaches&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
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@@ -299,22 +392,56 @@
   
 
 
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+  <title>Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems | FS</title>
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@@ -582,6 +788,7 @@ <h1>Search</h1>
 
 
 
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@@ -625,6 +832,7 @@ <h1>Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Sy
   
   
   
+  
 
   
   
@@ -637,15 +845,6 @@ <h1>Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Sy
 
 
 
-
-
-
-
-
-
-
-
-
   
 </div>
 
@@ -655,8 +854,7 @@ <h1>Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Sy
 
     <div class="article-style">
       <p>In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &ldquo;butterfly effect&rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.</p>
-<p>Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> <sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> <sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> (LSS) or non-intrusive least squares shadowing<sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup> (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl" target="_blank" rel="noopener">DiffEqSensitivity.jl</a> package.</p>
+<p>Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control<sup id="fnref:1"><a href="#fn:1" class="footnote-ref" role="doc-noteref">1</a></sup> <sup id="fnref:2"><a href="#fn:2" class="footnote-ref" role="doc-noteref">2</a></sup> <sup id="fnref:3"><a href="#fn:3" class="footnote-ref" role="doc-noteref">3</a></sup>. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> (LSS) or non-intrusive least squares shadowing<sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup> (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the <a href="https://github.com/SciML/DiffEqSensitivity.jl" target="_blank" rel="noopener">DiffEqSensitivity.jl</a> package.</p>
 <p>The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:</p>
 <p>$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -670,58 +868,51 @@ <h2 id="the-lorenz-system">The Lorenz system</h2>
 <p>One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:</p>
 <p>$$
 \begin{pmatrix}
-\text{d}x \\<br>
-\text{d}y \\<br>
-\text{d}z \\<br>
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\<br>
-x(\rho-z) - y\\<br>
-x y - \beta z \\<br>
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$</p>
 <p>For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:</p>
-<pre><code class="language-julia">using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
-
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
-
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &quot;initial&quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&quot;x&quot;, ylabel = L&quot;y&quot;, zlabel = L&quot;z&quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&quot;attractor&quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &quot;Lorenz_forward.png&quot;)
-</code></pre>
-
-
-
-
-
-  
-
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Random</span><span class="p">;</span> <span class="n">Random</span><span class="o">.</span><span class="n">seed!</span><span class="p">(</span><span class="mi">1234</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">OrdinaryDiffEq</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Statistics</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">QuadGK</span><span class="p">,</span> <span class="n">ForwardDiff</span><span class="p">,</span> <span class="n">Calculus</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">DiffEqSensitivity</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">SparseArrays</span><span class="p">,</span> <span class="n">LinearAlgebra</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># simulate 1 trajectory of the Lorenz system forward</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">lorenz!</span><span class="p">(</span><span class="n">du</span><span class="p">,</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">10</span><span class="o">*</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">3</span><span class="p">])</span> <span class="o">-</span> <span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">  <span class="n">du</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">-</span> <span class="p">(</span><span class="mi">8</span><span class="o">//</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">u</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="mf">28.0</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="n">tspan_init</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">30.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">tspan_attractor</span> <span class="o">=</span> <span class="p">(</span><span class="mf">30.0</span><span class="p">,</span><span class="mf">50.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">u0</span> <span class="o">=</span> <span class="n">rand</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob_init</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">lorenz!</span><span class="p">,</span><span class="n">u0</span><span class="p">,</span><span class="n">tspan_init</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol_init</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob_init</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">())</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob_attractor</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">lorenz!</span><span class="p">,</span><span class="n">sol_init</span><span class="p">[</span><span class="k">end</span><span class="p">],</span><span class="n">tspan_attractor</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol_attractor</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob_attractor</span><span class="p">,</span><span class="n">Vern9</span><span class="p">(),</span><span class="n">abstol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">,</span><span class="n">reltol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Plots</span><span class="p">,</span> <span class="n">LaTeXStrings</span>
+</span></span><span class="line"><span class="cl"><span class="n">pl1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol_init</span><span class="p">,</span><span class="n">vars</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">legend</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">label</span> <span class="o">=</span> <span class="s">&#34;initial&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">xlabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;x&#34;</span><span class="p">,</span> <span class="n">ylabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;y&#34;</span><span class="p">,</span> <span class="n">zlabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;z&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span><span class="mi">30</span><span class="p">),</span><span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">30</span><span class="p">,</span><span class="mi">30</span><span class="p">),</span><span class="n">zlims</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">49</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"> <span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">plot!</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span> <span class="n">sol_attractor</span><span class="p">,</span><span class="n">vars</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#34;attractor&#34;</span><span class="p">,</span><span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">25</span><span class="p">,</span><span class="mi">30</span><span class="p">),</span><span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">30</span><span class="p">,</span><span class="mi">30</span><span class="p">),</span><span class="n">zlims</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">49</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"> <span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">savefig</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span> <span class="s">&#34;Lorenz_forward.png&#34;</span><span class="p">)</span>
+</span></span></code></pre></div>
 
 
 
@@ -732,82 +923,79 @@ <h2 id="the-lorenz-system">The Lorenz system</h2>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/Lorenz_forward.png" >
 
 
-  <img src="/img/Lorenz_forward.png" alt=""  >
-</a>
 
 
 
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="Lorenz_forward.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.</p>
-<p>Following Refs.<sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> and <sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>, we choose</p>
+<p>Following Refs.<sup id="fnref1:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> and <sup id="fnref1:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>, we choose</p>
 <p>$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$</p>
 <p>as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.</p>
-<pre><code class="language-julia">function compute_objective(sol)
-  quadgk(t-&gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&quot;z(t)&quot;, xlabel=L&quot;t&quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&quot;\langle z\rangle&quot;, lw = 2)
-savefig(pl2, &quot;zsingle.png&quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">compute_objective</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">quadgk</span><span class="p">(</span><span class="n">t</span><span class="o">-&gt;</span> <span class="n">sol</span><span class="p">(</span><span class="n">t</span><span class="p">)[</span><span class="k">end</span><span class="p">]</span><span class="o">/</span><span class="p">(</span><span class="n">tspan_attractor</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">tspan_attractor</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="p">,</span><span class="n">tspan_attractor</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="n">tspan_attractor</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">atol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-10</span><span class="p">)[</span><span class="mi">1</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">pl2</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol_attractor</span><span class="o">.</span><span class="n">t</span><span class="p">,</span> <span class="n">getindex</span><span class="o">.</span><span class="p">(</span><span class="n">sol_attractor</span><span class="o">.</span><span class="n">u</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">ylabel</span><span class="o">=</span><span class="sa">L</span><span class="s">&#34;z(t)&#34;</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="sa">L</span><span class="s">&#34;t&#34;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span><span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">mean_z</span> <span class="o">=</span> <span class="p">[</span><span class="n">mean</span><span class="p">(</span><span class="n">getindex</span><span class="o">.</span><span class="p">(</span><span class="n">sol_attractor</span><span class="o">.</span><span class="n">u</span><span class="p">,</span><span class="mi">3</span><span class="p">))]</span>
+</span></span><span class="line"><span class="cl"><span class="n">int_z</span> <span class="o">=</span> <span class="n">compute_objective</span><span class="p">(</span><span class="n">sol_attractor</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">hline!</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span> <span class="p">[</span><span class="n">int_z</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="sa">L</span><span class="s">&#34;\langle z</span><span class="se">\r</span><span class="s">angle&#34;</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">savefig</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span> <span class="s">&#34;zsingle.png&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># for each value of the parameter, solve 20 times the initial value problem</span>
+</span></span><span class="line"><span class="cl"><span class="c"># wrap the procedure inside a function depending on p</span>
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">Lorenz_solve</span><span class="p">(</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">u0</span> <span class="o">=</span> <span class="n">rand</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">prob_init</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">lorenz!</span><span class="p">,</span><span class="n">u0</span><span class="p">,</span><span class="n">tspan_init</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">sol_init</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob_init</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">())</span>
+</span></span><span class="line"><span class="cl">  <span class="n">prob_attractor</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">lorenz!</span><span class="p">,</span><span class="n">sol_init</span><span class="p">[</span><span class="k">end</span><span class="p">],</span><span class="n">tspan_attractor</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">sol_attractor</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob_attractor</span><span class="p">,</span><span class="n">Vern9</span><span class="p">(),</span><span class="n">abstol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">,</span><span class="n">reltol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">sol_attractor</span><span class="p">,</span> <span class="n">prob_attractor</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">Niter</span> <span class="o">=</span> <span class="mi">10</span>
+</span></span><span class="line"><span class="cl"><span class="n">ps</span> <span class="o">=</span> <span class="n">collect</span><span class="p">(</span><span class="mf">0.0</span><span class="o">:</span><span class="mf">1.0</span><span class="o">:</span><span class="mf">50.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">probs</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl"><span class="n">sols</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl"><span class="n">zmean</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl"><span class="n">zstd</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl"><span class="k">for</span> <span class="n">ρ</span> <span class="k">in</span> <span class="n">ps</span>
+</span></span><span class="line"><span class="cl">  <span class="nd">@show</span> <span class="n">ρ</span>
+</span></span><span class="line"><span class="cl">  <span class="n">ztmp</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl">  <span class="k">for</span> <span class="n">i</span><span class="o">=</span><span class="mi">1</span><span class="o">:</span><span class="n">Niter</span>
+</span></span><span class="line"><span class="cl">    <span class="n">sol</span><span class="p">,</span> <span class="n">prob</span> <span class="o">=</span> <span class="n">Lorenz_solve</span><span class="p">([</span><span class="n">ρ</span><span class="p">])</span>
+</span></span><span class="line"><span class="cl">    <span class="n">zbar</span> <span class="o">=</span> <span class="n">compute_objective</span><span class="p">(</span><span class="n">sol</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">push!</span><span class="p">(</span><span class="n">sols</span><span class="p">,</span> <span class="n">sol</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">push!</span><span class="p">(</span><span class="n">probs</span><span class="p">,</span> <span class="n">prob</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">    <span class="n">push!</span><span class="p">(</span><span class="n">ztmp</span><span class="p">,</span> <span class="n">zbar</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="k">end</span>
+</span></span><span class="line"><span class="cl">  <span class="n">push!</span><span class="p">(</span><span class="n">zmean</span><span class="p">,</span><span class="n">mean</span><span class="p">(</span><span class="n">ztmp</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">  <span class="n">push!</span><span class="p">(</span><span class="n">zstd</span><span class="p">,</span><span class="n">std</span><span class="p">(</span><span class="n">ztmp</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">pl3</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">ps</span><span class="p">,</span><span class="n">zmean</span><span class="p">,</span> <span class="n">ribbon</span> <span class="o">=</span> <span class="n">zstd</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="sa">L</span><span class="s">&#34;\langle z</span><span class="se">\r</span><span class="s">angle&#34;</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="sa">L</span><span class="s">&#34;</span><span class="se">\r</span><span class="s">ho&#34;</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="nb">false</span><span class="p">,</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">50</span><span class="p">),</span><span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">50</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">savefig</span><span class="p">(</span><span class="n">pl3</span><span class="p">,</span> <span class="s">&#34;zvsrho.png&#34;</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">pl4</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">pl2</span><span class="p">,</span><span class="n">pl3</span><span class="p">,</span> <span class="n">margin</span><span class="o">=</span><span class="mi">3</span><span class="n">Plots</span><span class="o">.</span><span class="n">mm</span><span class="p">,</span> <span class="n">layout</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">600</span><span class="p">,</span><span class="mi">300</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">savefig</span><span class="p">(</span><span class="n">pl4</span><span class="p">,</span> <span class="s">&#34;z.png&#34;</span><span class="p">)</span>
+</span></span></code></pre></div><p>We obtain:</p>
 
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
 
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&quot;\langle z\rangle&quot;, xlabel=L&quot;\rho&quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &quot;zvsrho.png&quot;)
 
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &quot;z.png&quot;)
-</code></pre>
-<p>We obtain:</p>
 
 
 
 
 
-  
-
-
-
 
 
 
@@ -816,40 +1004,32 @@ <h2 id="the-lorenz-system">The Lorenz system</h2>
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/z.png" >
-
-
-  <img src="/img/z.png" alt=""  >
-</a>
-
-
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="z.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of</p>
 <p>$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$</p>
 <h2 id="conventional-forward-mode-sensitivity-analysis-and-finite-differencing">Conventional forward-mode sensitivity analysis and finite-differencing</h2>
-<p>For non-chaotic systems, we would just use the 
-<a href="https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms" target="_blank" rel="noopener">standard discrete or continuous forward sensitivity methods</a> or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:</p>
-<pre><code class="language-julia">function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-</code></pre>
-<p>we find diverging values:</p>
+<p>For non-chaotic systems, we would just use the <a href="https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms" target="_blank" rel="noopener">standard discrete or continuous forward sensitivity methods</a> or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="k">function</span> <span class="n">G</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">prob</span><span class="o">=</span><span class="n">prob_attractor</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">tmp_prob</span> <span class="o">=</span> <span class="n">remake</span><span class="p">(</span><span class="n">prob</span><span class="p">,</span><span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">tmp_sol</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">tmp_prob</span><span class="p">,</span><span class="n">Vern9</span><span class="p">(),</span><span class="n">abstol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">,</span><span class="n">reltol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">res</span> <span class="o">=</span> <span class="n">compute_objective</span><span class="p">(</span><span class="n">tmp_sol</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="nd">@info</span> <span class="n">res</span>
+</span></span><span class="line"><span class="cl">  <span class="n">res</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl"><span class="n">sense_forward</span> <span class="o">=</span> <span class="n">ForwardDiff</span><span class="o">.</span><span class="n">gradient</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">sense_calculus</span> <span class="o">=</span> <span class="n">Calculus</span><span class="o">.</span><span class="n">gradient</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span></code></pre></div><p>we find diverging values:</p>
 <p>$$
 \begin{aligned}
-&amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\<br>
+&amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$</p>
@@ -857,44 +1037,37 @@ <h2 id="conventional-forward-mode-sensitivity-analysis-and-finite-differencing">
 <p>$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$</p>
-<p>Similarly, using 
-<a href="https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor" target="_blank" rel="noopener">uncertainty quantification</a> one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:</p>
-<pre><code class="language-julia">prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &quot;Float64&quot;, labelfontsize=20, lw = 2,
-     xlabel = L&quot;x&quot;, ylabel = L&quot;y&quot;, zlabel = L&quot;z&quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&quot;Float32&quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
-
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &quot;Float64&quot;, labelfontsize=20, lw = 2,
-  xlabel = L&quot;x&quot;, ylabel = L&quot;y&quot;, zlabel = L&quot;z&quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&quot;Float32&quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-
-savefig(pl1, &quot;Lorenz_Floats.png&quot;)
-</code></pre>
-
-
-
-
-
-  
+<p>Similarly, using <a href="https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor" target="_blank" rel="noopener">uncertainty quantification</a> one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="n">prob_attractor1</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">lorenz!</span><span class="p">,</span><span class="n">sol_init</span><span class="p">[</span><span class="k">end</span><span class="p">],(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">50.0</span><span class="p">),</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob_attractor2</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">lorenz!</span><span class="p">,</span><span class="n">convert</span><span class="o">.</span><span class="p">(</span><span class="kt">Float32</span><span class="p">,</span> <span class="n">sol_init</span><span class="p">[</span><span class="k">end</span><span class="p">]),(</span><span class="mf">0f0</span><span class="p">,</span> <span class="mf">50f0</span><span class="p">),</span><span class="n">convert</span><span class="o">.</span><span class="p">(</span><span class="kt">Float32</span><span class="p">,</span><span class="n">p</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol1</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob_attractor1</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">abstol</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span><span class="n">reltol</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="n">saveat</span><span class="o">=</span><span class="mf">0.01</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol2</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob_attractor2</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">(),</span><span class="n">abstol</span><span class="o">=</span><span class="mf">1f-6</span><span class="p">,</span><span class="n">reltol</span><span class="o">=</span><span class="mf">1f-6</span><span class="p">,</span> <span class="n">saveat</span><span class="o">=</span><span class="mf">0.01f0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">list_plots</span> <span class="o">=</span> <span class="p">[]</span>
+</span></span><span class="line"><span class="cl"><span class="n">t1</span> <span class="o">=</span> <span class="mf">0.0</span>
+</span></span><span class="line"><span class="cl"><span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="mi">1</span><span class="o">:</span><span class="mi">500</span>
+</span></span><span class="line"><span class="cl">  <span class="n">t2</span> <span class="o">=</span> <span class="n">i</span><span class="o">*</span><span class="mf">0.1</span>
+</span></span><span class="line"><span class="cl">  <span class="n">plt1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol1</span><span class="p">,</span> <span class="n">vars</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">tspan</span><span class="o">=</span><span class="p">(</span><span class="n">t1</span><span class="p">,</span><span class="n">t2</span><span class="p">),</span> <span class="n">denseplot</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">     <span class="n">label</span> <span class="o">=</span> <span class="s">&#34;Float64&#34;</span><span class="p">,</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">     <span class="n">xlabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;x&#34;</span><span class="p">,</span> <span class="n">ylabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;y&#34;</span><span class="p">,</span> <span class="n">zlabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;z&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">     <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span><span class="mi">25</span><span class="p">),</span><span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">28</span><span class="p">,</span><span class="mi">25</span><span class="p">),</span><span class="n">zlims</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">48</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">  <span class="n">plot!</span><span class="p">(</span><span class="n">plt1</span><span class="p">,</span> <span class="n">sol2</span><span class="p">,</span><span class="n">vars</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">tspan</span><span class="o">=</span><span class="p">(</span><span class="n">t1</span><span class="p">,</span><span class="n">t2</span><span class="p">),</span> <span class="n">denseplot</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s">&#34;Float32&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">        <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span><span class="mi">25</span><span class="p">),</span><span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">28</span><span class="p">,</span><span class="mi">25</span><span class="p">),</span><span class="n">zlims</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">48</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl">  <span class="n">push!</span><span class="p">(</span><span class="n">list_plots</span><span class="p">,</span> <span class="n">plt1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">anim</span> <span class="o">=</span> <span class="n">animate</span><span class="p">(</span><span class="n">list_plots</span><span class="p">,</span><span class="n">every</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">pl1</span> <span class="o">=</span> <span class="n">plot</span><span class="p">(</span><span class="n">sol1</span><span class="p">,</span><span class="n">vars</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">legend</span><span class="o">=</span><span class="nb">true</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">label</span> <span class="o">=</span> <span class="s">&#34;Float64&#34;</span><span class="p">,</span> <span class="n">labelfontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">lw</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">xlabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;x&#34;</span><span class="p">,</span> <span class="n">ylabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;y&#34;</span><span class="p">,</span> <span class="n">zlabel</span> <span class="o">=</span> <span class="sa">L</span><span class="s">&#34;z&#34;</span><span class="p">,</span>
+</span></span><span class="line"><span class="cl">  <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span><span class="mi">25</span><span class="p">),</span><span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">28</span><span class="p">,</span><span class="mi">25</span><span class="p">),</span><span class="n">zlims</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">48</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"> <span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">plot!</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span> <span class="n">sol2</span><span class="p">,</span><span class="n">vars</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s">&#34;Float32&#34;</span><span class="p">,</span> <span class="n">xlims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">20</span><span class="p">,</span><span class="mi">25</span><span class="p">),</span><span class="n">ylims</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">28</span><span class="p">,</span><span class="mi">25</span><span class="p">),</span><span class="n">zlims</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">48</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"> <span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">savefig</span><span class="p">(</span><span class="n">pl1</span><span class="p">,</span> <span class="s">&#34;Lorenz_Floats.png&#34;</span><span class="p">)</span>
+</span></span></code></pre></div>
 
 
 
@@ -906,18 +1079,16 @@ <h2 id="conventional-forward-mode-sensitivity-analysis-and-finite-differencing">
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/Lorenz.gif" >
 
 
-  <img src="/img/Lorenz.gif" alt=""  >
-</a>
 
 
-
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="Lorenz.gif" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
 <p>Without animation:</p>
 
@@ -925,72 +1096,61 @@ <h2 id="conventional-forward-mode-sensitivity-analysis-and-finite-differencing">
 
 
 
-  
-
-
-
-
-
 
 
 
 
 
 
-<figure >
 
 
-  <a data-fancybox="" href="/img/Lorenz_Floats.png" >
 
 
-  <img src="/img/Lorenz_Floats.png" alt=""  >
-</a>
 
 
 
-</figure>
+<figure  >
+  <div class="d-flex justify-content-center">
+    <div class="w-100" ><img src="Lorenz_Floats.png" alt="" loading="lazy" data-zoomable /></div>
+  </div></figure>
 
-<p>Luckily, the 
-<a href="https://mathworld.wolfram.com/ShadowingTheorem.html" target="_blank" rel="noopener">shadowing lemma</a> states:</p>
+<p>Luckily, the <a href="https://mathworld.wolfram.com/ShadowingTheorem.html" target="_blank" rel="noopener">shadowing lemma</a> states:</p>
 <blockquote>
 <p>Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&ldquo;shadows&rdquo;) the numerically computed one.</p>
 </blockquote>
 <h2 id="shadowing-methods">Shadowing methods</h2>
-<p>The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do <em>not</em> diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. <sup id="fnref:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> and <sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>, including a visualization of both effects in Fig. 1 of Ref. <sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>.</p>
+<p>The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do <em>not</em> diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. <sup id="fnref2:4"><a href="#fn:4" class="footnote-ref" role="doc-noteref">4</a></sup> and <sup id="fnref2:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>, including a visualization of both effects in Fig. 1 of Ref. <sup id="fnref3:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>.</p>
 <h2 id="lss-and-nilss-for-the-lorenz-system">LSS and NILSS for the Lorenz system</h2>
-<p>Switching to LSS or NILSS within the 
-<a href="https://sciml.ai/" target="_blank" rel="noopener">SciML</a> ecosystem is straightforward by either defining the associated LSS (<code>ForwardLSSProblem</code> or <code>AdjointLSSProblem</code>) or NILSS problem (<code>NILSSProblem</code>) type manually:</p>
-<pre><code class="language-julia"># objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-</code></pre>
-<p>or by setting the <code>sensealg=</code> kwarg in <code>solve()</code>:</p>
-<pre><code class="language-julia"># select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&gt;GLSS(p),p) # 0.9694728321500617
-</code></pre>
-<p>Note that we have implemented three different options for forward shadowing with <code>LSS()</code>:</p>
+<p>Switching to LSS or NILSS within the <a href="https://sciml.ai/" target="_blank" rel="noopener">SciML</a> ecosystem is straightforward by either defining the associated LSS (<code>ForwardLSSProblem</code> or <code>AdjointLSSProblem</code>) or NILSS problem (<code>NILSSProblem</code>) type manually:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># objective</span>
+</span></span><span class="line"><span class="cl"><span class="n">g</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">t</span><span class="p">)</span> <span class="o">=</span> <span class="n">u</span><span class="p">[</span><span class="k">end</span><span class="p">]</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c">####</span>
+</span></span><span class="line"><span class="cl"><span class="c"># LSS</span>
+</span></span><span class="line"><span class="cl"><span class="c">####</span>
+</span></span><span class="line"><span class="cl"><span class="n">lss_problem</span> <span class="o">=</span> <span class="n">ForwardLSSProblem</span><span class="p">(</span><span class="n">sol_attractor</span><span class="p">,</span> <span class="n">ForwardLSS</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="n">DiffEqSensitivity</span><span class="o">.</span><span class="n">CosWindowing</span><span class="p">()),</span> <span class="n">g</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@show</span> <span class="n">shadow_forward</span><span class="p">(</span><span class="n">lss_problem</span><span class="p">)</span> <span class="c"># 1.0095888187322035</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">lss_problem</span> <span class="o">=</span> <span class="n">ForwardLSSProblem</span><span class="p">(</span><span class="n">sol_attractor</span><span class="p">,</span> <span class="n">ForwardLSS</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="n">DiffEqSensitivity</span><span class="o">.</span><span class="n">Cos2Windowing</span><span class="p">()),</span> <span class="n">g</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@show</span> <span class="n">shadow_forward</span><span class="p">(</span><span class="n">lss_problem</span><span class="p">)</span> <span class="c"># 1.0343951385924328</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">lss_problem</span> <span class="o">=</span> <span class="n">ForwardLSSProblem</span><span class="p">(</span><span class="n">sol_attractor</span><span class="p">,</span> <span class="n">ForwardLSS</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">10.0</span><span class="p">),</span> <span class="n">g</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@show</span> <span class="n">shadow_forward</span><span class="p">(</span><span class="n">lss_problem</span><span class="p">)</span> <span class="c"># 1.0284286902740765</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">adjointlss_problem</span> <span class="o">=</span> <span class="n">AdjointLSSProblem</span><span class="p">(</span><span class="n">sol_attractor</span><span class="p">,</span> <span class="n">AdjointLSS</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">10.0</span><span class="p">),</span> <span class="n">g</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@show</span> <span class="n">shadow_adjoint</span><span class="p">(</span><span class="n">adjointlss_problem</span><span class="p">)</span> <span class="c"># 1.028428690274077</span>
+</span></span></code></pre></div><p>or by setting the <code>sensealg=</code> kwarg in <code>solve()</code>:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c"># select via sensealg in solve</span>
+</span></span><span class="line"><span class="cl"><span class="k">using</span> <span class="n">Zygote</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">function</span> <span class="n">GLSS</span><span class="p">(</span><span class="n">p</span><span class="p">;</span> <span class="n">sensealg</span><span class="o">=</span><span class="n">ForwardLSS</span><span class="p">(),</span> <span class="n">dt</span><span class="o">=</span><span class="mf">0.01</span><span class="p">,</span> <span class="n">g</span><span class="o">=</span><span class="nb">nothing</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">_prob</span> <span class="o">=</span> <span class="n">remake</span><span class="p">(</span><span class="n">prob_attractor</span><span class="p">,</span><span class="n">p</span><span class="o">=</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">_sol</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">_prob</span><span class="p">,</span><span class="n">Vern9</span><span class="p">(),</span><span class="n">abstol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">,</span><span class="n">reltol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">,</span><span class="n">saveat</span><span class="o">=</span><span class="n">dt</span><span class="p">,</span><span class="n">sensealg</span><span class="o">=</span><span class="n">sensealg</span><span class="p">,</span> <span class="n">g</span><span class="o">=</span><span class="n">g</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">  <span class="n">sum</span><span class="p">(</span><span class="n">getindex</span><span class="o">.</span><span class="p">(</span><span class="n">_sol</span><span class="o">.</span><span class="n">u</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+</span></span><span class="line"><span class="cl"><span class="k">end</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">dp1</span> <span class="o">=</span> <span class="n">Zygote</span><span class="o">.</span><span class="n">gradient</span><span class="p">((</span><span class="n">p</span><span class="p">)</span><span class="o">-&gt;</span><span class="n">GLSS</span><span class="p">(</span><span class="n">p</span><span class="p">),</span><span class="n">p</span><span class="p">)</span> <span class="c"># 0.9694728321500617</span>
+</span></span></code></pre></div><p>Note that we have implemented three different options for forward shadowing with <code>LSS()</code>:</p>
 <ul>
 <li><code>CosWindowing()</code> (default)</li>
 <li><code>Cos2Windowing()</code></li>
@@ -999,72 +1159,68 @@ <h2 id="lss-and-nilss-for-the-lorenz-system">LSS and NILSS for the Lorenz system
 <p>Additionally, an adjoint implementation <code>AdjointLSS()</code> is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that <code>AdjointLSS(alpha=10.0)</code> agrees perfectly with <code>ForwardLSS(alpha=10.0)</code>. In all cases considered, we find the expected sensitivity value of $\approx 1$.</p>
 <p>However, the use of <code>LSS()</code> is (typically) much more expensive than the use of <code>NILSS()</code>, because <code>LSS()</code> needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of <code>NILSS()</code> scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making <code>NILSS()</code> more efficient.</p>
 <p>In the <code>NILSS()</code> algorithm, the user can control the number of steps per segment as well as the number of segments.</p>
-<pre><code class="language-julia">####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-</code></pre>
-<p>If the number of segments is chosen too small, a warning is thrown:</p>
-<pre><code class="language-julia">nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-</code></pre>
-<p>In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper<sup id="fnref:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>.</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="c">####</span>
+</span></span><span class="line"><span class="cl"><span class="c"># NILSS</span>
+</span></span><span class="line"><span class="cl"><span class="c">####</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># make sure trajectory is fully on the attractor</span>
+</span></span><span class="line"><span class="cl"><span class="n">Random</span><span class="o">.</span><span class="n">seed!</span><span class="p">(</span><span class="mi">1234</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">tspan_init</span> <span class="o">=</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span><span class="mf">100.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">tspan_attractor</span> <span class="o">=</span> <span class="p">(</span><span class="mf">100.0</span><span class="p">,</span><span class="mf">120.0</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">u0</span> <span class="o">=</span> <span class="n">rand</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob_init</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">lorenz!</span><span class="p">,</span><span class="n">u0</span><span class="p">,</span><span class="n">tspan_init</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="n">sol_init</span> <span class="o">=</span> <span class="n">solve</span><span class="p">(</span><span class="n">prob_init</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">())</span>
+</span></span><span class="line"><span class="cl"><span class="n">prob_attractor</span> <span class="o">=</span> <span class="n">ODEProblem</span><span class="p">(</span><span class="n">lorenz!</span><span class="p">,</span><span class="n">sol_init</span><span class="p">[</span><span class="k">end</span><span class="p">],</span><span class="n">tspan_attractor</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">nseg</span> <span class="o">=</span> <span class="mi">100</span> <span class="c"># number of segments on time interval</span>
+</span></span><span class="line"><span class="cl"><span class="n">nstep</span> <span class="o">=</span> <span class="mi">2001</span> <span class="c"># number of steps on each segment</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">nilss_prob</span> <span class="o">=</span> <span class="n">NILSSProblem</span><span class="p">(</span><span class="n">prob_attractor</span><span class="p">,</span> <span class="n">NILSS</span><span class="p">(</span><span class="n">nseg</span><span class="p">,</span> <span class="n">nstep</span><span class="p">),</span> <span class="n">g</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@show</span> <span class="n">DiffEqSensitivity</span><span class="o">.</span><span class="n">shadow_forward</span><span class="p">(</span><span class="n">nilss_prob</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">())</span> <span class="c"># 0.9966924374966089</span>
+</span></span></code></pre></div><p>If the number of segments is chosen too small, a warning is thrown:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-julia" data-lang="julia"><span class="line"><span class="cl"><span class="n">nseg</span> <span class="o">=</span> <span class="mi">20</span> <span class="c"># number of segments on time interval</span>
+</span></span><span class="line"><span class="cl"><span class="n">nstep</span> <span class="o">=</span> <span class="mi">2001</span> <span class="c"># number of steps on each segment</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="n">nilss_prob</span> <span class="o">=</span> <span class="n">NILSSProblem</span><span class="p">(</span><span class="n">prob_attractor</span><span class="p">,</span> <span class="n">NILSS</span><span class="p">(</span><span class="n">nseg</span><span class="p">,</span> <span class="n">nstep</span><span class="p">),</span> <span class="n">g</span><span class="p">)</span>
+</span></span><span class="line"><span class="cl"><span class="nd">@show</span> <span class="n">DiffEqSensitivity</span><span class="o">.</span><span class="n">shadow_forward</span><span class="p">(</span><span class="n">nilss_prob</span><span class="p">,</span><span class="n">Tsit5</span><span class="p">())</span> <span class="c"># 1.0416028730638789</span>
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="c"># Warning: Detected a large value of ξ at the beginning of a segment.</span>
+</span></span><span class="line"><span class="cl"><span class="c"># └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474</span>
+</span></span></code></pre></div><p>In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper<sup id="fnref4:5"><a href="#fn:5" class="footnote-ref" role="doc-noteref">5</a></sup>.</p>
 <h2 id="outlook">Outlook</h2>
 <p>With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as</p>
 <ul>
 <li>NILSAS<sup id="fnref:6"><a href="#fn:6" class="footnote-ref" role="doc-noteref">6</a></sup></li>
 <li>FD-NILSS<sup id="fnref:7"><a href="#fn:7" class="footnote-ref" role="doc-noteref">7</a></sup></li>
 </ul>
-<p>in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-<a href="https://github.com/SciML/DiffEqSensitivity.jl/issues/102" target="_blank" rel="noopener">issue</a> in the DiffEqSensitivity.jl package.</p>
+<p>in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding <a href="https://github.com/SciML/DiffEqSensitivity.jl/issues/102" target="_blank" rel="noopener">issue</a> in the DiffEqSensitivity.jl package.</p>
 <p>If you have any questions or comments, please don’t hesitate to contact me!</p>
-<section class="footnotes" role="doc-endnotes">
+<div class="footnotes" role="doc-endnotes">
 <hr>
 <ol>
-<li id="fn:1" role="doc-endnote">
-<p>Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. <strong>1</strong>, 035009 (2020). <a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:1">
+<p>Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. <strong>1</strong>, 035009 (2020).&#160;<a href="#fnref:1" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:2" role="doc-endnote">
-<p>Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. <strong>2</strong>, 035004 (2021). <a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:2">
+<p>Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. <strong>2</strong>, 035004 (2021).&#160;<a href="#fnref:2" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:3" role="doc-endnote">
-<p>Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). <a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:3">
+<p>Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&#160;<a href="#fnref:3" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:4" role="doc-endnote">
-<p>Qiqi Wang, Rui Hu, et al. J. Comput. Phys <strong>26</strong>, 210-224 (2014) <a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:4">
+<p>Qiqi Wang, Rui Hu, et al. J. Comput. Phys <strong>26</strong>, 210-224 (2014)&#160;<a href="#fnref:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref2:4" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:5" role="doc-endnote">
-<p>Angxiu Ni and Qiqi Wang. J. Comput. Phys <strong>347</strong>,  56-77 (2017). <a href="#fnref:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:5">
+<p>Angxiu Ni and Qiqi Wang. J. Comput. Phys <strong>347</strong>,  56-77 (2017).&#160;<a href="#fnref:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref1:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref2:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref3:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a>&#160;<a href="#fnref4:5" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:6" role="doc-endnote">
-<p>Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys <strong>395</strong>, 690-709, (2019) <a href="#fnref:6" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:6">
+<p>Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys <strong>395</strong>, 690-709, (2019)&#160;<a href="#fnref:6" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
-<li id="fn:7" role="doc-endnote">
-<p>Angxiu Ni, Qiqi Wang et al., J. Comput. Phys <strong>394</strong>, 615-631 (2019) <a href="#fnref:7" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
+<li id="fn:7">
+<p>Angxiu Ni, Qiqi Wang et al., J. Comput. Phys <strong>394</strong>, 615-631 (2019)&#160;<a href="#fnref:7" class="footnote-backref" role="doc-backlink">&#x21a9;&#xfe0e;</a></p>
 </li>
 </ol>
-</section>
+</div>
 
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@@ -1078,21 +1234,21 @@ <h2 id="outlook">Outlook</h2>
   
   <a class="badge badge-light" href="/tag/gsoc-2021/">GSoC 2021</a>
   
-  <a class="badge badge-light" href="/tag/julia/">julia</a>
+  <a class="badge badge-light" href="/tag/julia/">Julia</a>
   
-  <a class="badge badge-light" href="/tag/adjoint-sensitivity-methods/">Adjoint sensitivity methods</a>
+  <a class="badge badge-light" href="/tag/adjoint-sensitivity-methods/">Adjoint Sensitivity Methods</a>
   
-  <a class="badge badge-light" href="/tag/forward-sensitivity-methods/">Forward sensitivity methods</a>
+  <a class="badge badge-light" href="/tag/forward-sensitivity-methods/">Forward Sensitivity Methods</a>
   
   <a class="badge badge-light" href="/tag/shadowing/">Shadowing</a>
   
-  <a class="badge badge-light" href="/tag/chaotic-systems/">Chaotic systems</a>
+  <a class="badge badge-light" href="/tag/chaotic-systems/">Chaotic Systems</a>
   
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+  <div class="media author-card content-widget-hr">
     
       
+      <a href="https://frankschae.github.io/"><img class="avatar mr-3 avatar-circle" src="/authors/admin/avatar_hu5bdd272e648badb42a595cb37d3ba435_2457515_270x270_fill_lanczos_center_3.png" alt="Frank Schäfer"></a>
     
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-        <h6 class="card-subtitle">Postdoctoral researcher</h6>
-        <p class="card-text">My research interests include many-body physics, machine learning, and differentiable programming.</p>
-        <ul class="network-icon" aria-hidden="true">
+    <div class="media-body">
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+      <h6 class="card-subtitle">Postdoctoral researcher</h6>
+      <p class="card-text">My research interests include many-body physics, probabilistic machine learning, and differentiable programming.</p>
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+  <title>Performance Bounds for Quantum Control | FS</title>
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 <div class="article-container pt-3">
@@ -612,8 +821,11 @@ <h1>Performance Bounds for Quantum Control</h1>
   <div>
     
 
-  
-  <span>FS in collaboration with Flemming Holtorf</span>, <span>Julian Arnold</span>, <span>Chris Rackauckas</span>, <span>and Alan Edelman</span>
+  <span >
+      FS in Collaboration With Flemming Holtorf</span>, <span >
+      Julian Arnold</span>, <span >
+      Chris Rackauckas</span>, <span >
+      And Alan Edelman</span>
   </div>
   
   
@@ -635,6 +847,7 @@ <h1>Performance Bounds for Quantum Control</h1>
   
   
   
+  
 
   
   
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-    
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-  </a>
+  <a class="btn btn-outline-primary btn-page-header" href="https://arxiv.org/abs/2304.03366" target="_blank" rel="noopener">
+    arXiv preprint</a>
 
   
   
@@ -712,10 +914,8 @@ <h1>Performance Bounds for Quantum Control</h1>
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1" href="https://ieeexplore.ieee.org/abstract/document/10313626" target="_blank" rel="noopener">
-    
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-  </a>
+  <a class="btn btn-outline-primary btn-page-header" href="https://ieeexplore.ieee.org/abstract/document/10313626" target="_blank" rel="noopener">
+    QCE</a>
 
 
 </div>
@@ -726,7 +926,7 @@ <h1>Performance Bounds for Quantum Control</h1>
 
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+    <img src="/project/controlbounds/featured_hu9a224a6f0b76f74a72bd3d29668f06be_15378_15a2da1e4d2a73f61bf62118749f5df5.webp" width="373" height="226" alt="" class="featured-image">
     
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@@ -748,19 +948,19 @@ <h1>Performance Bounds for Quantum Control</h1>
 
 <div class="article-tags">
   
-  <a class="badge badge-light" href="/tag/quantum-control/">quantum control</a>
+  <a class="badge badge-light" href="/tag/quantum-control/">Quantum Control</a>
   
-  <a class="badge badge-light" href="/tag/convex-optimization/">convex optimization</a>
+  <a class="badge badge-light" href="/tag/convex-optimization/">Convex Optimization</a>
   
-  <a class="badge badge-light" href="/tag/stochastic-optimal-control/">stochastic optimal control</a>
+  <a class="badge badge-light" href="/tag/stochastic-optimal-control/">Stochastic Optimal Control</a>
   
-  <a class="badge badge-light" href="/tag/sum-of-squares/">sum-of-squares</a>
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@@ -835,8 +1057,12 @@ <h1>Performance Bounds for Quantum Control</h1>
       
       
       
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@@ -854,52 +1080,34 @@ <h1>Performance Bounds for Quantum Control</h1>
 
 
 
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@@ -614,8 +821,12 @@ <h1>Control of (Stochastic) Quantum Dynamics with Differentiable Programming</h1
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-  <span>FS in collaboration with Pavel Sekatski</span>, <span>Martin Koppenhöfer</span>, <span>Niels Lörch</span>, <span>Christoph Bruder</span>, <span>and Michal Kloc</span>
+  <span >
+      FS in Collaboration With Pavel Sekatski</span>, <span >
+      Martin Koppenhöfer</span>, <span >
+      Niels Lörch</span>, <span >
+      Christoph Bruder</span>, <span >
+      And Michal Kloc</span>
   </div>
   
   
@@ -637,6 +848,7 @@ <h1>Control of (Stochastic) Quantum Dynamics with Differentiable Programming</h1
   
   
   
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@@ -706,10 +909,8 @@ <h1>Control of (Stochastic) Quantum Dynamics with Differentiable Programming</h1
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
-    
-    SDE control paper
-  </a>
+  <a class="btn btn-outline-primary btn-page-header" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
+    SDE control paper</a>
 
   
   
@@ -722,10 +923,8 @@ <h1>Control of (Stochastic) Quantum Dynamics with Differentiable Programming</h1
   
     
   
-  <a class="btn btn-outline-primary my-1 mr-1" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
-    
-    ODE control paper
-  </a>
+  <a class="btn btn-outline-primary btn-page-header" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
+    ODE control paper</a>
 
   
   
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-    
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+  <a class="btn btn-outline-primary btn-page-header" href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=4" target="_blank" rel="noopener">
+    SIAM CSE 2021 talk</a>
 
   
   
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-    
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+  <a class="btn btn-outline-primary btn-page-header" href="https://www.youtube.com/watch?v=v8mJVGVdkNQ&amp;list=PLWIVj90xdDE-2eeyFuiooxWcF8kw323Iv&amp;index=3&amp;t=0s" target="_blank" rel="noopener">
+    CMD2020GEFES talk</a>
 
 
 </div>
@@ -768,7 +963,7 @@ <h1>Control of (Stochastic) Quantum Dynamics with Differentiable Programming</h1
 
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   <a class="badge badge-light" href="/tag/sciml/">SciML</a>
   
-  <a class="badge badge-light" href="/tag/differentiable-programming/">differentiable programming</a>
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   <a class="badge badge-light" href="/tag/nsde/">NSDE</a>
   
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   <a class="badge badge-light" href="/tag/automatic-differentiation/">Automatic Differentiation</a>
   
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-    Powered by the
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-    
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-        <span class="button_icon">
-          <i class="fas fa-chevron-up fa-2x"></i>
-        </span>
-      </a>
-    </span>
-    
-  </p>
-</footer>
 
-  </div>
+
+
+
+
+
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+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
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+
+  
   
 
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+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
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   <div class="modal-dialog">
@@ -1145,7 +1402,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -1160,5 +1418,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
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 </html>
diff --git a/project/index.html b/project/index.html
index 69d02a8..cc16bfd 100644
--- a/project/index.html
+++ b/project/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
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-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
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+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
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-  
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-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/project/">
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+
 
   
 
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+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
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+    
 
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@@ -94,6 +109,7 @@
       
     
       
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+      
 
       
       
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+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +141,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
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+      
       
 
       
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-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
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+  
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+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
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+
 
 
 
 
-  
 
 
   
   
-
   
-  <link rel="alternate" href="/project/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
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+
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+
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+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
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+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/project/" />
+<link rel="canonical" href="https://frankschae.github.io/project/" />
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+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
+
+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
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+
+<meta name="theme-color" content="#1565c0" />
+
+
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+
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+
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+
 
-  <link rel="canonical" href="https://frankschae.github.io/project/">
 
   
   
+
+
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+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
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+
+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
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+
   
+    <meta property="og:updated_time" content="2024-01-15T10:49:24-05:00" />
   
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+
   
   
-    
-    
   
+
   
-  <meta property="twitter:card" content="summary">
   
-  <meta property="og:site_name" content="FS">
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+    <link rel="alternate" href="/project/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2024-01-15T10:49:24-05:00">
+
   
 
+
   
+  <title>Projects | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>Projects | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   " data-wc-page-id="1b7f62207d2617f7491785ce1c71e57f" >
+
+  
+  
+  
+  
+  
+  
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+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
+
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+
+
+<aside class="search-modal" id="search">
   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
           <h1>Search</h1>
         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
+  
   
 
 
@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    <button type="button" class="navbar-toggler" data-toggle="collapse"
-            data-target="#navbar-content" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
-    <span><i class="fas fa-bars"></i></span>
-    </button>
-    
 
-    
-    <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    
-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
-        
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
 
-        
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
-        
-        
+        <ul class="navbar-nav d-md-inline-flex">
           
-        
 
-        
-        
-        
-        
-        
-        
           
+
           
           
-            
           
+            
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
+
+          
+          
 
-        
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-        
           
-        
 
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-        
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-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
-        
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+          
 
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-        
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+
           
           
           
-        
+            
+          
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#software"><span>Open Source Software</span></a>
-        </li>
+          
 
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+          
+          
 
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+          
+            
+              
+              
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+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
+          
+          
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+          
+
+          
+          
+          
+            
+          
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+          
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+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
 
         
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
@@ -461,63 +681,916 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
+
         
         
-          
-          
-          
-            
-          
-          
         
-
-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
+            </a>
+          </div>
         </li>
+        
 
         
         
 
-      
+      </ul>
+
+    </div>
+  </nav>
+</header>
+
+
+  </div>
+
+  <div class="page-body">
+    
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+  
+
+  
+  
+  
+    
+  
+<div class="universal-wrapper pt-3">
+  <h1>Projects</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  
+
+
+  
+
+  
+  
+    
+
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+  
+  
+  
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+
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+<div class="view-list view-list-item">
+  <i class="fas fa-project-diagram pub-icon" aria-hidden="true"></i>
+  <a href="/project/stochasticad/" >Automatic Differentiation of Programs with Discrete Randomness</a>
+
+  
+
+  
+  <div class="article-metadata">
+    
+
+  <span >
+      FS in Collaboration With Gaurav Arya</span>, <span >
+      Moritz Schauer</span>, <span >
+      Ruben Seyer</span>, <span >
+      Alex Lew</span>, <span >
+      Mathieu Huot</span>, <span >
+      Kartik Chandra</span>, <span >
+      Vikash Mansinghka</span>, <span >
+      Jonathan Ragan-Kelley</span>, <span >
+      Chris Rackauckas</span>
+  </div>
+  
+
+  
+  <div class="btn-links">
+    
+
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+  
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+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/gaurav-arya/StochasticAD.jl" target="_blank" rel="noopener">
+  Code
+</a>
+
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+  
+  
+  
+    
+  
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+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2210.08572" target="_blank" rel="noopener">
+    NeurIPS 2022</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2306.07961" target="_blank" rel="noopener">
+    ICML 2023 Workshop</a>
+
+
+  </div>
+  
+
+</div>
+
+  
+    
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+  
+  
+  
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+
+
+<div class="view-list view-list-item">
+  <i class="fas fa-project-diagram pub-icon" aria-hidden="true"></i>
+  <a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a>
+
+  
+
+  
+  <div class="article-metadata">
+    
+
+  <span >
+      FS in Collaboration With Flemming Holtorf</span>, <span >
+      Julian Arnold</span>, <span >
+      Chris Rackauckas</span>, <span >
+      And Alan Edelman</span>
+  </div>
+  
+
+  
+  <div class="btn-links">
+    
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+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2304.03366" target="_blank" rel="noopener">
+    arXiv preprint</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://ieeexplore.ieee.org/abstract/document/10313626" target="_blank" rel="noopener">
+    QCE</a>
+
+
+  </div>
+  
+
+</div>
+
+  
+    
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+  
+  
+  
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+
+
+<div class="view-list view-list-item">
+  <i class="fas fa-project-diagram pub-icon" aria-hidden="true"></i>
+  <a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a>
+
+  
+
+  
+  <div class="article-metadata">
+    
+
+  <span >
+      FS in Collaboration With Pavel Sekatski</span>, <span >
+      Martin Koppenhöfer</span>, <span >
+      Niels Lörch</span>, <span >
+      Christoph Bruder</span>, <span >
+      And Michal Kloc</span>
+  </div>
+  
+
+  
+  <div class="btn-links">
+    
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+  
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+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/frankschae/Control-of-Stochastic-Quantum-Dynamics-with-Differentiable-Programming" target="_blank" rel="noopener">
+  Code
+</a>
+
+
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+
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+  
+  
+  
+    
+  
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+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
+    SDE control paper</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
+    ODE control paper</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=4" target="_blank" rel="noopener">
+    SIAM CSE 2021 talk</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=v8mJVGVdkNQ&amp;list=PLWIVj90xdDE-2eeyFuiooxWcF8kw323Iv&amp;index=3&amp;t=0s" target="_blank" rel="noopener">
+    CMD2020GEFES talk</a>
+
+
+  </div>
+  
+
+</div>
+
+  
+    
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+  
+  
+  
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+
+
+<div class="view-list view-list-item">
+  <i class="fas fa-project-diagram pub-icon" aria-hidden="true"></i>
+  <a href="/project/reconstruction/" >Automated Data-driven Reconstruction of Physical Properties</a>
+
+  
+
+  
+  <div class="article-metadata">
+    
+
+  <span >
+      FS in Collaboration With Eliska Greplova</span>, <span >
+      Tomoki Hirosawa</span>, <span >
+      Jozef Bucko</span>, <span >
+      František Herman</span>, <span >
+      Rebekka Garreis</span>, <span >
+      Chuyao Tong</span>, <span >
+      Annika Kurzmann</span>, <span >
+      Thomas Ihn</span>, <span >
+      Hideaki Maebashi</span>, <span >
+      Hiroyasu Matsuura</span>, <span >
+      And Masao Ogata</span>
+  </div>
+  
+
+  
+  <div class="btn-links">
+    
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+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.19.024015" target="_blank" rel="noopener">
+    Phys. Rev. Applied</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.jps.jp/doi/abs/10.7566/JPSJ.91.114603" target="_blank" rel="noopener">
+    J. Phys. Soc. Jpn.</a>
+
+
+  </div>
+  
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+</div>
+
+  
+    
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+  
+  
+  
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+
+
+<div class="view-list view-list-item">
+  <i class="fas fa-project-diagram pub-icon" aria-hidden="true"></i>
+  <a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a>
+
+  
+
+  
+  <div class="article-metadata">
+    
+
+  <span >
+      FS in Collaboration With Julian Arnold</span>, <span >
+      Flemming Holtorf</span>, <span >
+      Eliska Greplova</span>, <span >
+      Agnes Valenti</span>, <span >
+      Martin Zonda</span>, <span >
+      Axel Lode</span>, <span >
+      Gregor Boschung</span>, <span >
+      Sebastian Huber</span>, <span >
+      And Niels Lörch</span>
+  </div>
+  
+
+  
+  <div class="btn-links">
+    
+
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+
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+
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+  
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+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/arnoldjulian/Replacing-neural-networks-by-optimal-analytical-predictors-for-the-detection-of-phase-transitions" target="_blank" rel="noopener">
+  Code
+</a>
+
+
+
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+
+
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+
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
+    PRE</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/1367-2630/ab7771/meta" target="_blank" rel="noopener">
+    NJP</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.033052" target="_blank" rel="noopener">
+    PRResearch</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.12.031044" target="_blank" rel="noopener">
+    PRX</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2311.09128" target="_blank" rel="noopener">
+    NeurIPS Workshop</a>
+
+
+  </div>
+  
+
+</div>
+
+  
+    
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+
+
+
+
+
+
+
+  
+  
+  
+
+
+
+
+<div class="view-list view-list-item">
+  <i class="fas fa-project-diagram pub-icon" aria-hidden="true"></i>
+  <a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a>
+
+  
+
+  
+  <div class="article-metadata">
+    
+
+  <span >
+      FS in Collaboration With Laurent De Forges De Parny</span>, <span >
+      Miguel Bastarrachea</span>, <span >
+      Axel Lode</span>, <span >
+      And Andreas Buchleitner</span>
+  </div>
+  
+
+  
+  <div class="btn-links">
+    
+
+
+
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+  
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+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.mdpi.com/1099-4300/22/4/382" target="_blank" rel="noopener">
+    Entropy</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://publikationen.uni-tuebingen.de/xmlui/handle/10900/87664" target="_blank" rel="noopener">
+    bwHPC proceeding</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://freidok.uni-freiburg.de/data/154730" target="_blank" rel="noopener">
+    Master thesis</a>
+
+
+  </div>
+  
+
+</div>
+
+  
+    
+
+
+
+
+
+
+
+
+
+  
+  
+  
+
+
+
+
+<div class="view-list view-list-item">
+  <i class="fas fa-project-diagram pub-icon" aria-hidden="true"></i>
+  <a href="/project/scattering/" >Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers</a>
+
+  
+
+  
+  <div class="article-metadata">
+    
+
+  <span >
+      FS in Collaboration With Felix Eckert and Thomas Wellens</span>
+  </div>
+  
+
+  
+  <div class="btn-links">
+    
+
+
+
+
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+
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+    
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/1361-6455/aa92d8/meta" target="_blank" rel="noopener">
+  PDF
+</a>
+
+
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+  
+  
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+    
+  
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+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://freidok.uni-freiburg.de/data/12809" target="_blank" rel="noopener">
+    Bachelor thesis</a>
+
 
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-    
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-    <div>
-      <h2><a href="/project/stochasticad/" >Automatic Differentiation of Programs with Discrete Randomness</a></h2>
-      <div class="article-style">
-        
-          We develop and implement AD algorithms for handling programs that can contain discrete randomness.
-        
-      </div>
-    </div>
-  
-    
-    
-    
-    <div>
-      <h2><a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a></h2>
-      <div class="article-style">
-        
-          We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
-        
-      </div>
-    </div>
   
-    
-    
-    
-    <div>
-      <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a></h2>
-      <div class="article-style">
-        
-          Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
-        
-      </div>
-    </div>
+
   
-    
-    
-    
-    <div>
-      <h2><a href="/project/reconstruction/" >Automated Data-driven Reconstruction of Physical Properties</a></h2>
-      <div class="article-style">
-        
-          We address the challenges of connecting measurements of a physical system to an underlying theoretical description.
-        
-      </div>
-    </div>
   
     
-    
-    
-    <div>
-      <h2><a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a></h2>
-      <div class="article-style">
-        
-          Data-driven methods based on sample instances of the state of a physical system as a function of the system's parameters.
-        
-      </div>
-    </div>
   
-    
-    
-    
-    <div>
-      <h2><a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a></h2>
-      <div class="article-style">
-        
-          Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
-        
-      </div>
-    </div>
   
     
-    
-    
-    <div>
-      <h2><a href="/project/scattering/" >Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers</a></h2>
-      <div class="article-style">
-        
-          Numerical optimization of the positions of point scatterers to maximize the total scattering cross section for an incoming plane wave.
-        
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@@ -714,42 +1718,18 @@ <h2><a href="/project/scattering/" >Cooperative Scattering of Scalar Waves by Op
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@@ -612,8 +821,11 @@ <h1>Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double We
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-  <span>FS in collaboration with Laurent de Forges de Parny</span>, <span>Miguel Bastarrachea</span>, <span>Axel Lode</span>, <span>and Andreas Buchleitner</span>
+  <span >
+      FS in Collaboration With Laurent De Forges De Parny</span>, <span >
+      Miguel Bastarrachea</span>, <span >
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@@ -635,6 +847,7 @@ <h1>Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double We
   
   
   
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+  <a class="btn btn-outline-primary btn-page-header" href="https://www.mdpi.com/1099-4300/22/4/382" target="_blank" rel="noopener">
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+    bwHPC proceeding</a>
 
   
   
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@@ -764,11 +962,11 @@ <h1>Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double We
 
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-  <a class="badge badge-light" href="/tag/many-body-physics/">Many-body physics</a>
+  <a class="badge badge-light" href="/tag/many-body-physics/">Many-Body Physics</a>
   
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-  
-  <span>FS in collaboration with Julian Arnold</span>, <span>Flemming Holtorf</span>, <span>Eliska Greplova</span>, <span>Agnes Valenti</span>, <span>Martin Zonda</span>, <span>Axel Lode</span>, <span>Gregor Boschung</span>, <span>Sebastian Huber</span>, <span>and Niels Lörch</span>
+  <span >
+      FS in Collaboration With Julian Arnold</span>, <span >
+      Flemming Holtorf</span>, <span >
+      Eliska Greplova</span>, <span >
+      Agnes Valenti</span>, <span >
+      Martin Zonda</span>, <span >
+      Axel Lode</span>, <span >
+      Gregor Boschung</span>, <span >
+      Sebastian Huber</span>, <span >
+      And Niels Lörch</span>
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+<!DOCTYPE html>
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+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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@@ -614,8 +821,18 @@ <h1>Automated Data-driven Reconstruction of Physical Properties</h1>
   <div>
     
 
-  
-  <span>FS in collaboration with Eliska Greplova</span>, <span>Tomoki Hirosawa</span>, <span>Jozef Bucko</span>, <span>František Herman</span>, <span>Rebekka Garreis</span>, <span>Chuyao Tong</span>, <span>Annika Kurzmann</span>, <span>Thomas Ihn</span>, <span>Hideaki Maebashi</span>, <span>Hiroyasu Matsuura</span>, <span>and Masao Ogata</span>
+  <span >
+      FS in Collaboration With Eliska Greplova</span>, <span >
+      Tomoki Hirosawa</span>, <span >
+      Jozef Bucko</span>, <span >
+      František Herman</span>, <span >
+      Rebekka Garreis</span>, <span >
+      Chuyao Tong</span>, <span >
+      Annika Kurzmann</span>, <span >
+      Thomas Ihn</span>, <span >
+      Hideaki Maebashi</span>, <span >
+      Hiroyasu Matsuura</span>, <span >
+      And Masao Ogata</span>
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@@ -637,6 +854,7 @@ <h1>Automated Data-driven Reconstruction of Physical Properties</h1>
   
   
   
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+  <a class="btn btn-outline-primary btn-page-header" href="https://journals.jps.jp/doi/abs/10.7566/JPSJ.91.114603" target="_blank" rel="noopener">
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@@ -298,22 +386,56 @@
   
 
 
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+  <title>Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers | FS</title>
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-  <title>Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers | FS</title>
 
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@@ -610,8 +814,8 @@ <h1>Cooperative Scattering of Scalar Waves by Optimized Configurations of Point
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+  <span >
+      FS in Collaboration With Felix Eckert and Thomas Wellens</span>
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@@ -621,6 +825,8 @@ <h1>Cooperative Scattering of Scalar Waves by Optimized Configurations of Point
     
     
       
+          Last updated on
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@@ -633,6 +839,7 @@ <h1>Cooperative Scattering of Scalar Waves by Optimized Configurations of Point
   
   
   
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 </div>
@@ -714,7 +910,7 @@ <h1>Cooperative Scattering of Scalar Waves by Optimized Configurations of Point
 
 <div class="article-header article-container featured-image-wrapper mt-4 mb-4" style="max-width: 600px; max-height: 400px;">
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@@ -752,8 +948,12 @@ <h1>Cooperative Scattering of Scalar Waves by Optimized Configurations of Point
       
       
       
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   "author": {
     "@type": "Person",
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@@ -300,22 +393,56 @@
   
 
 
+
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+  <title>Automatic Differentiation of Programs with Discrete Randomness | FS</title>
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+
 
 
 
 
 
-  <title>Automatic Differentiation of Programs with Discrete Randomness | FS</title>
 
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 <div class="article-container pt-3">
@@ -612,8 +821,16 @@ <h1>Automatic Differentiation of Programs with Discrete Randomness</h1>
   <div>
     
 
-  
-  <span>FS in collaboration with Gaurav Arya</span>, <span>Moritz Schauer</span>, <span>Ruben Seyer</span>, <span>Alex Lew</span>, <span>Mathieu Huot</span>, <span>Kartik Chandra</span>, <span>Vikash Mansinghka</span>, <span>Jonathan Ragan-Kelley</span>, <span>Chris Rackauckas</span>
+  <span >
+      FS in Collaboration With Gaurav Arya</span>, <span >
+      Moritz Schauer</span>, <span >
+      Ruben Seyer</span>, <span >
+      Alex Lew</span>, <span >
+      Mathieu Huot</span>, <span >
+      Kartik Chandra</span>, <span >
+      Vikash Mansinghka</span>, <span >
+      Jonathan Ragan-Kelley</span>, <span >
+      Chris Rackauckas</span>
   </div>
   
   
@@ -635,6 +852,7 @@ <h1>Automatic Differentiation of Programs with Discrete Randomness</h1>
   
   
   
+  
 
   
   
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-    
-    NeurIPS 2022
-  </a>
+  <a class="btn btn-outline-primary btn-page-header" href="https://arxiv.org/abs/2210.08572" target="_blank" rel="noopener">
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-  <a class="btn btn-outline-primary my-1 mr-1" href="https://arxiv.org/abs/2306.07961" target="_blank" rel="noopener">
-    
-    ICML 2023 Workshop
-  </a>
+  <a class="btn btn-outline-primary btn-page-header" href="https://arxiv.org/abs/2306.07961" target="_blank" rel="noopener">
+    ICML 2023 Workshop</a>
 
 
 </div>
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@@ -792,8 +1005,14 @@ <h1>Automatic Differentiation of Programs with Discrete Randomness</h1>
       
       
       
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@@ -1100,32 +1325,54 @@ <h1>Automatic Differentiation of Programs with Discrete Randomness</h1>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
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+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
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@@ -1137,7 +1384,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
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         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
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diff --git a/publication-type/2/index.html b/publication-type/2/index.html
index 594567d..b8f0a9b 100644
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+++ b/publication-type/2/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
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+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
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-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
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         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
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+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
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+    </div>
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-      <div class="article-style">
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-          We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact …
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+  
   
 
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         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
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         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -774,5 +1362,25 @@ <h5 class="modal-title">Cite</h5>
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diff --git a/publication-type/2/index.xml b/publication-type/2/index.xml
index 2591b11..9a88ffc 100644
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+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sun, 10 May 2020 13:22:24 +0200</lastBuildDate>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
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index 8b004fc..5ad132a 100644
--- a/publication-type/2/page/1/index.html
+++ b/publication-type/2/page/1/index.html
@@ -1 +1,10 @@
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\ No newline at end of file
+<!DOCTYPE html>
+<html lang="en-us">
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+    <title>https://frankschae.github.io/publication-type/2/</title>
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diff --git a/publication-type/article-journal/index.html b/publication-type/article-journal/index.html
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+          <h1>Search</h1>
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+        <div class="col-6 col-search-close">
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
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+        <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
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+      <div id="search-hits">
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+      <button type="button" class="navbar-toggler" data-toggle="collapse"
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+            <a class="nav-link " href="/#about"><span>Home</span></a>
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+          
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+              
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+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
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+
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+          
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+          
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+          
+            
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+              
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+                
+              
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+            
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+
+          <li class="nav-item">
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+
+    
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+
+  
+  
+  
+    
+  
+<div class="universal-wrapper pt-3">
+  <h1>Article-Journal</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  
+
+
+  
+
+  
+  
+    
+
+
+
+
+
+
+
+  
+
+
+
+
+
+
+
+  
+
+
+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/publication/journal-article/" >An example journal article</a>
+    </div>
+
+    
+    <a href="/publication/journal-article/"  class="summary-link">
+      <div class="article-style">
+        Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+      <div>
+        
+
+  <span class="author-highlighted">
+      Frank Schäfer</span><i class="author-notes fas fa-info-circle" data-toggle="tooltip" title="Equal contribution"></i>, <span >
+      Robert Ford</span><i class="author-notes fas fa-info-circle" data-toggle="tooltip" title="Equal contribution"></i>
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+
+
+
+<a href="#" class="btn btn-outline-primary btn-page-header btn-sm js-cite-modal"
+        data-filename="/publication/journal-article/cite.bib">
+  Cite
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+
+  
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
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+
+
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+
+
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+
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+            alt="An example journal article" loading="lazy">
+      </a>
+    
+  </div>
+</div>
+
+  
+
+  
+
+  
+
+</div>
+  </div>
+
+  <div class="page-footer">
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+    
+    <div class="container">
+      <footer class="site-footer">
+
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+
+
+
+
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+
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+
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+  
+
+  
+  
+    
+  
+  
+    
+  
+
+  
+
+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
+  
+
+  <p class="powered-by footer-license-icons">
+    <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank" aria-label="Creative Commons">
+      <i class="fab fa-creative-commons fa-2x" aria-hidden="true"></i>
+      <i class="fab fa-creative-commons-by fa-2x" aria-hidden="true"></i>
+      
+        <i class="fab fa-creative-commons-nc fa-2x" aria-hidden="true"></i>
+      
+      
+        <i class="fab fa-creative-commons-nd fa-2x" aria-hidden="true"></i>
+      
+    </a>
+  </p>
+
+
+
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+
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new file mode 100644
index 0000000..3cbb334
--- /dev/null
+++ b/publication-type/article-journal/index.xml
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+  <channel>
+    <title>Article-Journal | FS</title>
+    <link>https://frankschae.github.io/publication-type/article-journal/</link>
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+      <pubDate>Tue, 01 Sep 2015 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/journal-article/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
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+  &lt;/div&gt;
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+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
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diff --git a/publication-type/article-journal/page/1/index.html b/publication-type/article-journal/page/1/index.html
new file mode 100644
index 0000000..43ca1aa
--- /dev/null
+++ b/publication-type/article-journal/page/1/index.html
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+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/publication-type/article-journal/</title>
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diff --git a/publication-type/article/index.html b/publication-type/article/index.html
new file mode 100644
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+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
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+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
+        
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+            <a href="#" class="dropdown-item js-set-theme-dark">
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+            <a href="#" class="dropdown-item js-set-theme-auto">
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+    
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+
+  
+
+  
+  
+  
+    
+  
+<div class="universal-wrapper pt-3">
+  <h1>Article</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  
+
+
+  
+
+  
+  
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+
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+
+
+
+
+  
+
+
+
+
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+  
+
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/publication/preprint/" >An example preprint / working paper</a>
+    </div>
+
+    
+    <a href="/publication/preprint/"  class="summary-link">
+      <div class="article-style">
+        Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+      <div>
+        
+
+  <span class="author-highlighted">
+      Frank Schäfer</span>
+      </div>
+      
+    </div>
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new file mode 100644
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+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
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+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
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+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
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+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
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+              <span>Light</span>
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+  </nav>
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+    
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+
+
+
+
+
+  
+
+  
+  
+  
+    
+  
+<div class="universal-wrapper pt-3">
+  <h1>Paper-Conference</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  
+
+
+  
+
+  
+  
+    
+
+
+
+
+
+
+
+  
+
+
+
+
+
+
+
+  
+
+
+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/publication/conference-paper/" >An example conference paper</a>
+    </div>
+
+    
+    <a href="/publication/conference-paper/"  class="summary-link">
+      <div class="article-style">
+        Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+      <div>
+        
+
+  <span class="author-highlighted">
+      Frank Schäfer</span><i class="author-notes fas fa-info-circle" data-toggle="tooltip" title="Equal contribution"></i>, <span >
+      Robert Ford</span><i class="author-notes fas fa-info-circle" data-toggle="tooltip" title="Equal contribution"></i>
+      </div>
+      
+    </div>
+
+    
+    <div class="btn-links">
+      
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+  
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+
+
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="/publication/conference-paper/conference-paper.pdf" target="_blank" rel="noopener">
+  PDF
+</a>
+
+
+
+<a href="#" class="btn btn-outline-primary btn-page-header btn-sm js-cite-modal"
+        data-filename="/publication/conference-paper/cite.bib">
+  Cite
+</a>
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
+  Dataset
+</a>
+
+
+
+  
+    
+  
+
+
+
+
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="/slides/example/" target="_blank">
+    Slides
+  </a>
+  
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://youtube.com" target="_blank" rel="noopener">
+  Video
+</a>
+
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+  
+  
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
+  Source Document
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+    </div>
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+    
+    
+      
+      
+      <a href="/publication/conference-paper/" >
+        <img src="/publication/conference-paper/featured_hu3d03a01dcc18bc5be0e67db3d8d209a6_312700_996bd74732f186a85f43ddbf53f3cbed.webp" height="100" width="150"
+            alt="An example conference paper" loading="lazy">
+      </a>
+    
+  </div>
+</div>
+
+  
+
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+</div>
+  </div>
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+      <footer class="site-footer">
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+  
+  
+  
+
+  
+  
+    
+  
+  
+    
+  
+
+  
+
+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
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+
+
+
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+  
+
+  
+  
+
+  
+
+
+
+
+
+
+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
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+
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+    <script src="https://cdn.jsdelivr.net/gh/julmot/mark.js@8.11.1/dist/jquery.mark.min.js" integrity="sha512-mhbv5DqBMgrWL+32MmsDOt/OAvqr/cHimk6B8y/bx/xS88MVkYGPiVv2ixKVrkywF2qHplNRUvFsAHUdxZ3Krg==" crossorigin="anonymous"></script>
+  
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+  
+  
+  
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+  
+  
+
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
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+
+
+  
+<div id="modal" class="modal fade" role="dialog">
+  <div class="modal-dialog">
+    <div class="modal-content">
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+        <h5 class="modal-title">Cite</h5>
+        <button type="button" class="close" data-dismiss="modal" aria-label="Close">
+          <span aria-hidden="true">&times;</span>
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+
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+</body>
+</html>
diff --git a/publication-type/paper-conference/index.xml b/publication-type/paper-conference/index.xml
new file mode 100644
index 0000000..7a4e44a
--- /dev/null
+++ b/publication-type/paper-conference/index.xml
@@ -0,0 +1,35 @@
+<?xml version="1.0" encoding="utf-8" standalone="yes" ?>
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+  <channel>
+    <title>Paper-Conference | FS</title>
+    <link>https://frankschae.github.io/publication-type/paper-conference/</link>
+      <atom:link href="https://frankschae.github.io/publication-type/paper-conference/index.xml" rel="self" type="application/rss+xml" />
+    <description>Paper-Conference</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Mon, 01 Jul 2013 00:00:00 +0000</lastBuildDate>
+    <image>
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+      <title>Paper-Conference</title>
+      <link>https://frankschae.github.io/publication-type/paper-conference/</link>
+    </image>
+    
+    <item>
+      <title>An example conference paper</title>
+      <link>https://frankschae.github.io/publication/conference-paper/</link>
+      <pubDate>Mon, 01 Jul 2013 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/conference-paper/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Click the &lt;em&gt;Cite&lt;/em&gt; button above to demo the feature to enable visitors to import publication metadata into their reference management software.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
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+    
+  </channel>
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new file mode 100644
index 0000000..4aeca68
--- /dev/null
+++ b/publication-type/paper-conference/page/1/index.html
@@ -0,0 +1,10 @@
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/publication-type/paper-conference/</title>
+    <link rel="canonical" href="https://frankschae.github.io/publication-type/paper-conference/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
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diff --git a/publication/conference-paper/cite.bib b/publication/conference-paper/cite.bib
new file mode 100644
index 0000000..98c3748
--- /dev/null
+++ b/publication/conference-paper/cite.bib
@@ -0,0 +1,8 @@
+@inproceedings{example1,
+  title={An example conference paper},
+  author={Bighetti, Nelson and Ford, Robert},
+  booktitle={Source Themes Conference},
+  pages={1--6},
+  year={2013},
+  organization={IEEE}
+}
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+  
+  
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+
+
+<div class="article-container pt-3">
+  <h1>An example conference paper</h1>
+
+  
+
+  
+
+
+<div class="article-metadata">
+
+  
+  
+  
+  
+  <div>
+    
+
+  <span class="author-highlighted">
+      Frank Schäfer</span><i class="author-notes fas fa-info-circle" data-toggle="tooltip" title="Equal contribution"></i>, <span >
+      Robert Ford</span><i class="author-notes fas fa-info-circle" data-toggle="tooltip" title="Equal contribution"></i>
+  </div>
+  
+  
+
+  
+  <span class="article-date">
+    
+    
+      
+    
+    July, 2013
+  </span>
+  
+
+  
+
+  
+
+  
+  
+  
+  
+
+  
+  
+
+</div>
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+<div class="btn-links mb-3">
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+<a class="btn btn-outline-primary btn-page-header" href="/publication/conference-paper/conference-paper.pdf" target="_blank" rel="noopener">
+  PDF
+</a>
+
+
+
+<a href="#" class="btn btn-outline-primary btn-page-header js-cite-modal"
+        data-filename="/publication/conference-paper/cite.bib">
+  Cite
+</a>
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
+  Code
+</a>
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
+  Dataset
+</a>
+
+
+
+  
+    
+  
+
+
+
+
+  
+  <a class="btn btn-outline-primary btn-page-header" href="/slides/example/" target="_blank">
+    Slides
+  </a>
+  
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header" href="https://youtube.com" target="_blank" rel="noopener">
+  Video
+</a>
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
+  Source Document
+</a>
+
+
+
+
+</div>
+
+
+</div>
+
+
+<div class="article-header article-container featured-image-wrapper mt-4 mb-4" style="max-width: 720px; max-height: 480px;">
+  <div style="position: relative">
+    <img src="/publication/conference-paper/featured_hu3d03a01dcc18bc5be0e67db3d8d209a6_312700_c28b67e81f3f23a444f1b2b3b64df884.webp" width="720" height="480" alt="" class="featured-image">
+    <span class="article-header-caption">Image credit: <a href="https://unsplash.com/photos/pLCdAaMFLTE" target="_blank" rel="noopener"><strong>Unsplash</strong></a></span>
+  </div>
+</div>
+
+
+
+  <div class="article-container">
+
+    
+    <h3>Abstract</h3>
+    <p class="pub-abstract">Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum. Sed ac faucibus dolor, scelerisque sollicitudin nisi. Cras purus urna, suscipit quis sapien eu, pulvinar tempor diam. Quisque risus orci, mollis id ante sit amet, gravida egestas nisl. Sed ac tempus magna. Proin in dui enim. Donec condimentum, sem id dapibus fringilla, tellus enim condimentum arcu, nec volutpat est felis vel metus. Vestibulum sit amet erat at nulla eleifend gravida.</p>
+    
+
+    
+    
+    <div class="row">
+      <div class="col-md-1"></div>
+      <div class="col-md-10">
+        <div class="row">
+          <div class="col-12 col-md-3 pub-row-heading">Type</div>
+          <div class="col-12 col-md-9">
+            <a href="/publication/#paper-conference">
+              Conference paper
+            </a>
+          </div>
+        </div>
+      </div>
+      <div class="col-md-1"></div>
+    </div>
+    <div class="d-md-none space-below"></div>
+    
+
+    
+    <div class="row">
+      <div class="col-md-1"></div>
+      <div class="col-md-10">
+        <div class="row">
+          <div class="col-12 col-md-3 pub-row-heading">Publication</div>
+          <div class="col-12 col-md-9">In <em>Hugo Blox Builder Conference</em></div>
+        </div>
+      </div>
+      <div class="col-md-1"></div>
+    </div>
+    <div class="d-md-none space-below"></div>
+    
+
+    <div class="space-below"></div>
+
+    <div class="article-style"><div class="alert alert-note">
+  <div>
+    Click the <em>Cite</em> button above to demo the feature to enable visitors to import publication metadata into their reference management software.
+  </div>
+</div>
+<div class="alert alert-note">
+  <div>
+    Create your slides in Markdown - click the <em>Slides</em> button to check out the example.
+  </div>
+</div>
+<p>Add the publication&rsquo;s <strong>full text</strong> or <strong>supplementary notes</strong> here. You can use rich formatting such as including <a href="https://docs.hugoblox.com/content/writing-markdown-latex/" target="_blank" rel="noopener">code, math, and images</a>.</p>
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+  <div class="media author-card content-widget-hr">
+    
+      
+      <a href="https://frankschae.github.io/"><img class="avatar mr-3 avatar-circle" src="/authors/admin/avatar_hu5bdd272e648badb42a595cb37d3ba435_2457515_270x270_fill_lanczos_center_3.png" alt="Frank Schäfer"></a>
+    
+
+    <div class="media-body">
+      <h5 class="card-title"><a href="https://frankschae.github.io/">Frank Schäfer</a></h5>
+      <h6 class="card-subtitle">Postdoctoral researcher</h6>
+      <p class="card-text">My research interests include many-body physics, probabilistic machine learning, and differentiable programming.</p>
+      <ul class="network-icon" aria-hidden="true">
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+        <i class="ai ai-google-scholar"></i>
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+
   
 
   
@@ -605,7 +821,7 @@ <h1>F. Schäfer, M. A. Bastarrachea-Magnani, A. U. J. Lode, L. de Forges de Parn
     
       
     
-    May 2020
+    May, 2020
   </span>
   
 
@@ -616,22 +832,14 @@ <h1>F. Schäfer, M. A. Bastarrachea-Magnani, A. U. J. Lode, L. de Forges de Parn
   
   
   
+  
 
   
   
 
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@@ -653,7 +861,7 @@ <h1>F. Schäfer, M. A. Bastarrachea-Magnani, A. U. J. Lode, L. de Forges de Parn
 
 
 
-<a class="btn btn-outline-primary my-1 mr-1" href="https://www.mdpi.com/1099-4300/22/4/382" target="_blank" rel="noopener">
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   PDF
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-  <a class="badge badge-light" href="/tag/bosonic-systems/">Bosonic systems</a>
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-
   
   
 
 
 
-
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+  
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           <a href="{{relpermalink}}">{{title}}</a>
@@ -1044,42 +1334,18 @@ <h6 class="card-subtitle">Postdoctoral researcher</h6>
           <p class="search-hit-description">{{snippet}}</p>
         </div>
       </div>
-      </div>
-    </script>
-    
+    </div>
+  </script>
+  
+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
+    <script src="https://cdn.jsdelivr.net/gh/julmot/mark.js@8.11.1/dist/jquery.mark.min.js" integrity="sha512-mhbv5DqBMgrWL+32MmsDOt/OAvqr/cHimk6B8y/bx/xS88MVkYGPiVv2ixKVrkywF2qHplNRUvFsAHUdxZ3Krg==" crossorigin="anonymous"></script>
+  
 
-    
-    
-    <script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/3.2.1/fuse.min.js" integrity="sha256-VzgmKYmhsGNNN4Ph1kMW+BjoYJM2jV5i4IlFoeZA9XI=" crossorigin="anonymous"></script>
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-    
 
-    
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-    
-    
 
-    
-    
 
-    
-    
-    
-    
-    
-    
-    
-    
-    
-      
-    
-    
-    
-    
-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
-    
 
 
 
@@ -1088,32 +1354,54 @@ <h6 class="card-subtitle">Postdoctoral researcher</h6>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
-    <a href="https://gohugo.io" target="_blank" rel="noopener">Hugo</a>.
 
-    
-    <span class="float-right" aria-hidden="true">
-      <a href="#" class="back-to-top">
-        <span class="button_icon">
-          <i class="fas fa-chevron-up fa-2x"></i>
-        </span>
-      </a>
-    </span>
-    
-  </p>
-</footer>
 
-  </div>
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+  
   
 
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
+
   
 <div id="modal" class="modal fade" role="dialog">
   <div class="modal-dialog">
@@ -1125,7 +1413,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -1140,5 +1429,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
+
+
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+
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+
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+
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+
 </body>
 </html>
diff --git a/publication/index.html b/publication/index.html
index e0ee4ff..6e594c7 100644
--- a/publication/index.html
+++ b/publication/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/publication/">
+
+
+
+
+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
-  <script src="/js/mathjax-config.js"></script>
   
 
   
+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
+
   
   
   
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/academicons/1.8.6/css/academicons.min.css" integrity="sha256-uFVgMKfistnJAfoCUQigIl+JfUaP47GrRKjf6CTPVmw=" crossorigin="anonymous">
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-
     
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1.9.4/css/academicons.min.css" integrity="sha512-IW0nhlW5MgNydsXJO40En2EoCkTTjZhI3yuODrZIc8cQ4h1XcF53PsqDHa09NqnkXuIe0Oiyyj171BqZFwISBw==" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
+
     
     
-      
     
     
       
       
-        
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/github.min.css" crossorigin="anonymous" title="hl-light">
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/dracula.min.css" crossorigin="anonymous" title="hl-dark" disabled>
-        
-      
     
-
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.css" integrity="sha256-SHMGCYmST46SoyGgo4YR/9AlK1vf3ff84Aq9yK4hdqM=" crossorigin="anonymous">
     
 
     
-
     
     
-      
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +141,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
+  
+  
+  
+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
+
+  
+  
   
 
   
   
   
   
-  <link rel="stylesheet" href="/css/academic.css">
+  
+  
+  
+    
+    
+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
+  
 
   
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 
 
 
 
-  
 
 
   
   
-
   
-  <link rel="alternate" href="/publication/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
+
+
+
+
+
   
 
-  <link rel="manifest" href="/index.webmanifest">
-  <link rel="icon" type="image/png" href="/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_2.png">
-  <link rel="apple-touch-icon" type="image/png" href="/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_192x192_fill_lanczos_center_2.png">
+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
+
+
+
+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/publication/" />
+<link rel="canonical" href="https://frankschae.github.io/publication/" />
+
+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
+
+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
+<link rel="apple-touch-icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_180x180_fill_lanczos_center_3.png" />
+
+<meta name="theme-color" content="#1565c0" />
+
+
+
+
+
+
+
+
 
-  <link rel="canonical" href="https://frankschae.github.io/publication/">
 
   
   
+
+
+
+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
+<meta property="twitter:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" />
+
+
+
   
+
+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/publication/" />
+<meta property="og:title" content="Publications | FS" />
+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
+
   
+    <meta property="og:updated_time" content="2020-05-10T13:22:24&#43;02:00" />
   
+
+
+
+
+
+
+
+
+
+
   
   
-    
-    
   
+
   
-  <meta property="twitter:card" content="summary">
   
-  <meta property="og:site_name" content="FS">
-  <meta property="og:url" content="https://frankschae.github.io/publication/">
-  <meta property="og:title" content="Publications | FS">
-  <meta property="og:description" content="Postdoctoral researcher"><meta property="og:image" content="https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png">
-  <meta property="twitter:image" content="https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png"><meta property="og:locale" content="en-us">
+    <link rel="alternate" href="/publication/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2020-05-10T13:22:24&#43;02:00">
+
   
 
+
   
+  <title>Publications | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>Publications | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   " data-wc-page-id="3a079e7dad19be978a318345a7749d34" >
+
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
+
+  
+
+
+<aside class="search-modal" id="search">
   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
           <h1>Search</h1>
         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
+  
   
 
 
@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    <button type="button" class="navbar-toggler" data-toggle="collapse"
-            data-target="#navbar-content" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
-    <span><i class="fas fa-bars"></i></span>
-    </button>
-    
 
-    
-    <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    
-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
-        
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
 
-        
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
-        
-        
+        <ul class="navbar-nav d-md-inline-flex">
           
-        
 
-        
-        
-        
-        
-        
-        
           
+
           
           
-            
           
+            
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
-        
-        
-        
           
-        
+          
 
-        
-        
-        
-        
-        
-        
           
+
           
           
           
-        
+            
+          
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#software"><span>Open Source Software</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
+          
+          
+
+          
+
+          
+          
+          
+            
+          
+
+          
+
+          
+          
+          
+          
+
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
 
         
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
@@ -461,65 +681,54 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
+
         
         
-          
-          
-          
-            
-          
-          
         
-
-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
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+            <a href="#" class="dropdown-item js-set-theme-dark">
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+
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@@ -563,6 +772,15 @@ <h1>Publications</h1>
         
         
       
+        
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+      
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+          <input type="search" class="filter-search form-control form-control-sm" placeholder="Search..." autocapitalize="off"
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+              2
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+            <option value=".pubtype-paper-conference">
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@@ -596,6 +825,18 @@ <h1>Publications</h1>
               2020
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+            <option value=".year-2019">
+              2019
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+            
+            <option value=".year-2015">
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+            
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/1361-6455/aa92d8/meta" target="_blank" rel="noopener">
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+<div class="pub-list-item view-citation" style="margin-bottom: 1rem">
   <i class="far fa-file-alt pub-icon" aria-hidden="true"></i>
 
   
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-<a class="btn btn-outline-primary my-1 mr-1 btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/frankschae/Vector-field-divergence-of-predictive-model-output-as-indication-of-phase-transitions" target="_blank" rel="noopener">
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+      Frank Schäfer</span>
+  </span>
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+<a href="#" class="btn btn-outline-primary btn-page-header btn-sm js-cite-modal"
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@@ -1029,7 +1664,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -1044,5 +1680,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
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diff --git a/publication/index.xml b/publication/index.xml
index a97c6df..359a793 100644
--- a/publication/index.xml
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@@ -5,9 +5,9 @@
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     <description>Publications</description>
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+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sun, 10 May 2020 13:22:24 +0200</lastBuildDate>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
       <title>Publications</title>
       <link>https://frankschae.github.io/publication/</link>
     </image>
@@ -44,5 +44,57 @@
       <description></description>
     </item>
     
+    <item>
+      <title>An example preprint / working paper</title>
+      <link>https://frankschae.github.io/publication/preprint/</link>
+      <pubDate>Sun, 07 Apr 2019 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/preprint/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
+</description>
+    </item>
+    
+    <item>
+      <title>An example journal article</title>
+      <link>https://frankschae.github.io/publication/journal-article/</link>
+      <pubDate>Tue, 01 Sep 2015 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/journal-article/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Click the &lt;em&gt;Cite&lt;/em&gt; button above to demo the feature to enable visitors to import publication metadata into their reference management software.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
+</description>
+    </item>
+    
+    <item>
+      <title>An example conference paper</title>
+      <link>https://frankschae.github.io/publication/conference-paper/</link>
+      <pubDate>Mon, 01 Jul 2013 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/conference-paper/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Click the &lt;em&gt;Cite&lt;/em&gt; button above to demo the feature to enable visitors to import publication metadata into their reference management software.
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+    
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new file mode 100644
index 0000000..a7da7ac
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+++ b/publication/journal-article/cite.bib
@@ -0,0 +1,8 @@
+@article{example2,
+  title = {An example journal article},
+  author={Bighetti, Nelson and Ford, Robert},
+  journal = {Journal of Source Themes},
+  year    = 2015,
+  volume  = 1,
+  number  = 1
+}
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+            <a class="nav-link " href="/#about"><span>Home</span></a>
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diff --git a/publication/jphysb/index.html b/publication/jphysb/index.html
index 3a0531b..247e2b0 100644
--- a/publication/jphysb/index.html
+++ b/publication/jphysb/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
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+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
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+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
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+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering cross section for an incoming plane wave, on the one hand, and to minimize the decay rate associated to a long-lived scattering resonance, on the other. In both cases, the optimum is achieved by configurations where all scatterers are placed on a line parallel to the direction of the incoming plane wave. The associated maximal scattering cross section increases quadratically with the number of scatterers for large N, whereas the minimal decay rate—which is realized by configurations that are not the same as those that maximize the scattering cross section—decreases exponentially as a function of N. Finally, we also analyze the stability of our optimized configurations with respect to small random displacements of the scatterers. These results demonstrate that optimized configurations of scatterers bear a considerable potential for applications such as quantum memories or mirrors consisting of only a few atoms.">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/publication/jphysb/">
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   "description": "We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering cross section for an incoming plane wave, on the one hand, and to minimize the decay rate associated to a long-lived scattering resonance, on the other. In both cases, the optimum is achieved by configurations where all scatterers are placed on a line parallel to the direction of the incoming plane wave. The associated maximal scattering cross section increases quadratically with the number of scatterers for large N, whereas the minimal decay rate—which is realized by configurations that are not the same as those that maximize the scattering cross section—decreases exponentially as a function of N. Finally, we also analyze the stability of our optimized configurations with respect to small random displacements of the scatterers. These results demonstrate that optimized configurations of scatterers bear a considerable potential for applications such as quantum memories or mirrors consisting of only a few atoms."
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-<a class="btn btn-outline-primary my-1 mr-1" href="https://iopscience.iop.org/article/10.1088/1361-6455/aa92d8/meta" target="_blank" rel="noopener">
+<a class="btn btn-outline-primary btn-page-header" href="https://iopscience.iop.org/article/10.1088/1361-6455/aa92d8/meta" target="_blank" rel="noopener">
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 <div class="article-tags">
   
-  <a class="badge badge-light" href="/tag/multiple-scattering/">multiple scattering</a>
+  <a class="badge badge-light" href="/tag/multiple-scattering/">Multiple Scattering</a>
   
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+  <a class="badge badge-light" href="/tag/numerical-optimization/">Numerical Optimization</a>
   
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-        <p class="card-text">My research interests include many-body physics, machine learning, and differentiable programming.</p>
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+      <h5 class="card-title"><a href="https://frankschae.github.io/">Frank Schäfer</a></h5>
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+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
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@@ -1086,32 +1352,54 @@ <h6 class="card-subtitle">Postdoctoral researcher</h6>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
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-</footer>
 
-  </div>
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@@ -1123,7 +1411,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
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         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
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diff --git a/publication/njp/index.html b/publication/njp/index.html
index e040cf9..ebcbf56 100644
--- a/publication/njp/index.html
+++ b/publication/njp/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
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+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
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+
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+
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+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/publication/njp/">
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+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
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+  
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-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
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diff --git a/publication/pre99/index.html b/publication/pre99/index.html
index 5dcbcee..09d7b66 100644
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-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
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-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
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+
 
   
   
-  
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-  <meta name="description" content="We introduce an alternative method to identify phase boundaries in physical systems. It is based on training a predictive model such as a neural network to infer a physical system&#39;s parameters from its state. The deviation of the inferred parameters from the underlying correct parameters will be most susceptible and diverge maximally in the vicinity of phase boundaries. Therefore, peaks in the vector field divergence of the model&#39;s predictions are used as indication of phase transitions. Our method is applicable for phase diagrams of arbitrary parameter dimension and without prior information about the phases. Application to both the two-dimensional Ising model and the dissipative Kuramoto-Hopf model show promising results.">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/publication/pre99/">
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@@ -605,7 +821,7 @@ <h1>F. Schäfer and N. Lörch, Phys. Rev. E 99, 062107 (2019)</h1>
     
       
     
-    May 2020
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@@ -616,22 +832,14 @@ <h1>F. Schäfer and N. Lörch, Phys. Rev. E 99, 062107 (2019)</h1>
   
   
   
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+<a class="btn btn-outline-primary btn-page-header" href="https://github.com/frankschae/Vector-field-divergence-of-predictive-model-output-as-indication-of-phase-transitions" target="_blank" rel="noopener">
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@@ -697,18 +905,16 @@ <h3>Abstract</h3>
     
 
     
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 <div class="article-tags">
   
-  <a class="badge badge-light" href="/tag/machine-learning/">Machine learning</a>
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@@ -787,8 +1001,14 @@ <h3>Abstract</h3>
       
       
       
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+        <a href="https://www.linkedin.com/shareArticle?url=https%3A%2F%2Ffrankschae.github.io%2Fpublication%2Fpre99%2F&amp;title=F.&#43;Sch%C3%A4fer&#43;and&#43;N.&#43;L%C3%B6rch%2C&#43;Phys.&#43;Rev.&#43;E&#43;99%2C&#43;062107&#43;%282019%29" target="_blank" rel="noopener" class="share-btn-linkedin" aria-label="linkedin-in">
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+      
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+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+  
+  
+
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
+
+  
+<div id="modal" class="modal fade" role="dialog">
+  <div class="modal-dialog">
+    <div class="modal-content">
+      <div class="modal-header">
+        <h5 class="modal-title">Cite</h5>
+        <button type="button" class="close" data-dismiss="modal" aria-label="Close">
+          <span aria-hidden="true">&times;</span>
+        </button>
+      </div>
+      <div class="modal-body">
+        
+        <pre><code></code></pre>
+      </div>
+      <div class="modal-footer">
+        <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
+          <i class="fas fa-copy"></i> Copy
+        </a>
+        <a class="btn btn-outline-primary my-1 js-download-cite" href="#" target="_blank">
+          <i class="fas fa-download"></i> Download
+        </a>
+        <div id="modal-error"></div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
+
+
+
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+
+
+</body>
+</html>
diff --git a/publication_types/index.html b/publication_types/index.html
index d1b4cc9..372008f 100644
--- a/publication_types/index.html
+++ b/publication_types/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/publication_types/">
+
+
+
+
+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
-  <script src="/js/mathjax-config.js"></script>
   
 
   
+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
+
   
   
   
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/academicons/1.8.6/css/academicons.min.css" integrity="sha256-uFVgMKfistnJAfoCUQigIl+JfUaP47GrRKjf6CTPVmw=" crossorigin="anonymous">
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-
     
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1.9.4/css/academicons.min.css" integrity="sha512-IW0nhlW5MgNydsXJO40En2EoCkTTjZhI3yuODrZIc8cQ4h1XcF53PsqDHa09NqnkXuIe0Oiyyj171BqZFwISBw==" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
+
     
     
-      
     
     
       
       
-        
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/github.min.css" crossorigin="anonymous" title="hl-light">
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/dracula.min.css" crossorigin="anonymous" title="hl-dark" disabled>
-        
-      
     
-
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.css" integrity="sha256-SHMGCYmST46SoyGgo4YR/9AlK1vf3ff84Aq9yK4hdqM=" crossorigin="anonymous">
     
 
     
-
     
     
-      
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
+  
+  
+  
+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
+
+  
+  
   
 
   
   
   
   
-  <link rel="stylesheet" href="/css/academic.css">
+  
+  
+  
+    
+    
+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
+  
 
   
+  
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+
 
 
 
 
-  
 
 
   
   
-
   
-  <link rel="alternate" href="/publication_types/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
+
+
+
+
+
   
 
-  <link rel="manifest" href="/index.webmanifest">
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+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
+
+
+
+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/publication_types/" />
+<link rel="canonical" href="https://frankschae.github.io/publication_types/" />
+
+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
+
+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
+<link rel="apple-touch-icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_180x180_fill_lanczos_center_3.png" />
+
+<meta name="theme-color" content="#1565c0" />
+
+
+
+
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+
 
-  <link rel="canonical" href="https://frankschae.github.io/publication_types/">
 
   
   
+
+
+
+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
+<meta property="twitter:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" />
+
+
+
   
+
+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/publication_types/" />
+<meta property="og:title" content="Publication_types | FS" />
+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
+
   
+    <meta property="og:updated_time" content="2020-05-10T13:22:24&#43;02:00" />
   
+
+
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+
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+
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+
   
   
-    
-    
   
+
   
-  <meta property="twitter:card" content="summary">
   
-  <meta property="og:site_name" content="FS">
-  <meta property="og:url" content="https://frankschae.github.io/publication_types/">
-  <meta property="og:title" content="Publication_types | FS">
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+    <link rel="alternate" href="/publication_types/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2020-05-10T13:22:24&#43;02:00">
+
   
 
+
   
+  <title>Publication_types | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>Publication_types | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
+
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
+
+  
+
+
+<aside class="search-modal" id="search">
   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
           <h1>Search</h1>
         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
+  
   
 
 
@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    <button type="button" class="navbar-toggler" data-toggle="collapse"
-            data-target="#navbar-content" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
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-    </button>
-    
 
-    
-    <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    
-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
-        
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
 
-        
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
-        
-        
+        <ul class="navbar-nav d-md-inline-flex">
           
-        
 
-        
-        
-        
-        
-        
-        
           
+
           
           
-            
           
+            
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
-        
-        
-        
           
-        
+          
 
-        
-        
-        
-        
-        
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+
           
           
           
-        
+            
+          
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#software"><span>Open Source Software</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
+          
+          
+
+          
+
+          
+          
+          
+            
+          
+
+          
+
+          
+          
+          
+          
+
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
 
         
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
@@ -461,65 +681,51 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
+
         
         
-          
-          
-          
-            
-          
-          
         
-
-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
+            </a>
+          </div>
         </li>
+        
 
         
         
 
-      
-
-        
       </ul>
-    </div>
-
-    <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
-      
-      <li class="nav-item">
-        <a class="nav-link js-search" href="#"><i class="fas fa-search" aria-hidden="true"></i></a>
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-      
-
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-        <div class="dropdown-menu">
-          <a href="#" class="dropdown-item js-set-theme-light">
-            <span>Light</span>
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-          <a href="#" class="dropdown-item js-set-theme-auto">
-            <span>Automatic</span>
-          </a>
-        </div>
-      </li>
-      
 
-      
+    </div>
+  </nav>
+</header>
 
-    </ul>
 
   </div>
-</nav>
 
+  <div class="page-body">
+    
+    
+    
+
+    
 
-  
 
 
 
@@ -552,165 +758,516 @@ <h1>Publication_types</h1>
 <div class="universal-wrapper">
   
 
+
   
+
   
-    
-    
-    
-    <div>
-      <h2><a href="/publication-type/2/" >2</a></h2>
-      <div class="article-style">
-        
-      </div>
-    </div>
   
+    
 
-  
 
-</div>
 
-      
 
-    
-    
-    
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-  <url>
-    <loc>https://frankschae.github.io/author/frank-schafer/</loc>
-  </url>
-  
-  <url>
-    <loc>https://frankschae.github.io/author/fs-in-collaboration-with-felix-eckert-and-thomas-wellens/</loc>
-  </url>
-  
-  <url>
+    <changefreq>weekly</changefreq>
+  </url><url>
+    <loc>https://frankschae.github.io/home/</loc>
+    <changefreq>weekly</changefreq>
+  </url><url>
     <loc>https://frankschae.github.io/tag/multiple-scattering-theory/</loc>
+    <changefreq>weekly</changefreq>
   </url>
-  
-  <url>
-    <loc>https://frankschae.github.io/talk/</loc>
-  </url>
-  
 </urlset>
diff --git a/slides/example/index.html b/slides/example/index.html
new file mode 100644
index 0000000..b934c91
--- /dev/null
+++ b/slides/example/index.html
@@ -0,0 +1,321 @@
+<!DOCTYPE html>
+<html lang="en-us">
+<head>
+
+  
+  
+  
+  
+
+  <meta charset="utf-8">
+  <meta name="viewport" content="width=device-width, initial-scale=1">
+  <meta http-equiv="X-UA-Compatible" content="IE=edge">
+  <meta name="generator" content="Hugo Blox Builder 5.9.7">
+
+  
+    <link rel="manifest" href="/manifest.webmanifest">
+  
+
+  <link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png">
+  <link rel="apple-touch-icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_192x192_fill_lanczos_center_3.png">
+
+  <link rel="canonical" href="https://frankschae.github.io/slides/example/">
+
+  <title>Slides | FS</title>
+
+  
+  <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/reveal.js@4.6.1/dist/reveal.min.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/reveal.js@4.6.1/dist/theme/black.min.css">
+
+  
+  
+  
+  
+    
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css">
+  
+
+  
+  
+  
+</head>
+<body>
+
+  
+<div class="reveal">
+  <div class="slides">
+    
+    
+    
+
+    
+    
+    
+    
+    
+
+    
+    
+
+    
+    
+    
+    <section>
+    
+      <h1 id="create-slides-in-markdown-with-hugo-blox-builder">Create slides in Markdown with Hugo Blox Builder</h1>
+<p><a href="https://hugoblox.com/" target="_blank" rel="noopener">Hugo Blox Builder</a> | <a href="https://docs.hugoblox.com/content/slides/" target="_blank" rel="noopener">Documentation</a></p>
+
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h2 id="features">Features</h2>
+<ul>
+<li>Efficiently write slides in Markdown</li>
+<li>3-in-1: Create, Present, and Publish your slides</li>
+<li>Supports speaker notes</li>
+<li>Mobile friendly slides</li>
+</ul>
+
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h2 id="controls">Controls</h2>
+<ul>
+<li>Next: <code>Right Arrow</code> or <code>Space</code></li>
+<li>Previous: <code>Left Arrow</code></li>
+<li>Start: <code>Home</code></li>
+<li>Finish: <code>End</code></li>
+<li>Overview: <code>Esc</code></li>
+<li>Speaker notes: <code>S</code></li>
+<li>Fullscreen: <code>F</code></li>
+<li>Zoom: <code>Alt + Click</code></li>
+<li><a href="https://revealjs.com/pdf-export/" target="_blank" rel="noopener">PDF Export</a></li>
+</ul>
+
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h2 id="code-highlighting">Code Highlighting</h2>
+<p>Inline code: <code>variable</code></p>
+<p>Code block:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="cl"><span class="n">porridge</span> <span class="o">=</span> <span class="s2">&#34;blueberry&#34;</span>
+</span></span><span class="line"><span class="cl"><span class="k">if</span> <span class="n">porridge</span> <span class="o">==</span> <span class="s2">&#34;blueberry&#34;</span><span class="p">:</span>
+</span></span><span class="line"><span class="cl">    <span class="nb">print</span><span class="p">(</span><span class="s2">&#34;Eating...&#34;</span><span class="p">)</span>
+</span></span></code></pre></div>
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h2 id="math">Math</h2>
+<p>In-line math: $x + y = z$</p>
+<p>Block math:</p>
+<p>$$
+f\left( x \right) = ;\frac{{2\left( {x + 4} \right)\left( {x - 4} \right)}}{{\left( {x + 4} \right)\left( {x + 1} \right)}}
+$$</p>
+
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h2 id="fragments">Fragments</h2>
+<p>Make content appear incrementally</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-fallback" data-lang="fallback"><span class="line"><span class="cl">{{% fragment %}} One {{% /fragment %}}
+</span></span><span class="line"><span class="cl">{{% fragment %}} **Two** {{% /fragment %}}
+</span></span><span class="line"><span class="cl">{{% fragment %}} Three {{% /fragment %}}
+</span></span></code></pre></div><p>Press <code>Space</code> to play!</p>
+<span class="fragment " >
+  One
+</span>
+<span class="fragment " >
+  <strong>Two</strong>
+</span>
+<span class="fragment " >
+  Three
+</span>
+
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<p>A fragment can accept two optional parameters:</p>
+<ul>
+<li><code>class</code>: use a custom style (requires definition in custom CSS)</li>
+<li><code>weight</code>: sets the order in which a fragment appears</li>
+</ul>
+
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h2 id="speaker-notes">Speaker Notes</h2>
+<p>Add speaker notes to your presentation</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-markdown" data-lang="markdown"><span class="line"><span class="cl">{{% speaker_note %}}
+</span></span><span class="line"><span class="cl">
+</span></span><span class="line"><span class="cl"><span class="k">-</span> Only the speaker can read these notes
+</span></span><span class="line"><span class="cl"><span class="k">-</span> Press <span class="sb">`S`</span> key to view
+</span></span><span class="line"><span class="cl">  {{% /speaker_note %}}
+</span></span></code></pre></div><p>Press the <code>S</code> key to view the speaker notes!</p>
+<aside class="notes">
+  <ul>
+<li>Only the speaker can read these notes</li>
+<li>Press <code>S</code> key to view</li>
+</ul>
+
+</aside>
+
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h2 id="themes">Themes</h2>
+<ul>
+<li>black: Black background, white text, blue links (default)</li>
+<li>white: White background, black text, blue links</li>
+<li>league: Gray background, white text, blue links</li>
+<li>beige: Beige background, dark text, brown links</li>
+<li>sky: Blue background, thin dark text, blue links</li>
+</ul>
+
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<ul>
+<li>night: Black background, thick white text, orange links</li>
+<li>serif: Cappuccino background, gray text, brown links</li>
+<li>simple: White background, black text, blue links</li>
+<li>solarized: Cream-colored background, dark green text, blue links</li>
+</ul>
+
+    </section>
+    
+
+    
+    
+    
+      
+
+<section data-noprocess data-shortcode-slide
+  
+      
+      data-background-image="/media/boards.jpg"
+  >
+
+<h2 id="custom-slide">Custom Slide</h2>
+<p>Customize the slide style and background</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-markdown" data-lang="markdown"><span class="line"><span class="cl">{{<span class="p">&lt;</span> <span class="nt">slide</span> <span class="na">background-image</span><span class="o">=</span><span class="s">&#34;/media/boards.jpg&#34;</span> <span class="p">&gt;</span>}}
+</span></span><span class="line"><span class="cl">{{<span class="p">&lt;</span> <span class="nt">slide</span> <span class="na">background-color</span><span class="o">=</span><span class="s">&#34;#0000FF&#34;</span> <span class="p">&gt;</span>}}
+</span></span><span class="line"><span class="cl">{{<span class="p">&lt;</span> <span class="nt">slide</span> <span class="na">class</span><span class="o">=</span><span class="s">&#34;my-style&#34;</span> <span class="p">&gt;</span>}}
+</span></span></code></pre></div>
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h2 id="custom-css-example">Custom CSS Example</h2>
+<p>Let&rsquo;s make headers navy colored.</p>
+<p>Create <code>assets/css/reveal_custom.css</code> with:</p>
+<div class="highlight"><pre tabindex="0" class="chroma"><code class="language-css" data-lang="css"><span class="line"><span class="cl"><span class="p">.</span><span class="nc">reveal</span> <span class="nt">section</span> <span class="nt">h1</span><span class="o">,</span>
+</span></span><span class="line"><span class="cl"><span class="p">.</span><span class="nc">reveal</span> <span class="nt">section</span> <span class="nt">h2</span><span class="o">,</span>
+</span></span><span class="line"><span class="cl"><span class="p">.</span><span class="nc">reveal</span> <span class="nt">section</span> <span class="nt">h3</span> <span class="p">{</span>
+</span></span><span class="line"><span class="cl">  <span class="k">color</span><span class="p">:</span> <span class="kc">navy</span><span class="p">;</span>
+</span></span><span class="line"><span class="cl"><span class="p">}</span>
+</span></span></code></pre></div>
+    </section>
+    
+
+    
+    
+    
+    <section>
+    
+      
+<h1 id="questions">Questions?</h1>
+<p><a href="https://discord.gg/z8wNYzb" target="_blank" rel="noopener">Ask</a></p>
+<p><a href="https://docs.hugoblox.com/content/slides/" target="_blank" rel="noopener">Documentation</a></p>
+
+    </section>
+    
+
+    
+    
+  </div>
+</div>
+
+
+
+  
+  <script src="https://cdn.jsdelivr.net/npm/reveal.js@4.6.1/dist/reveal.min.js" crossorigin="anonymous"></script>
+  <script src="https://cdn.jsdelivr.net/npm/reveal.js@4.6.1/plugin/markdown/markdown.min.js" crossorigin="anonymous"></script>
+  <script src="https://cdn.jsdelivr.net/npm/reveal.js@4.6.1/plugin/notes/notes.min.js" crossorigin="anonymous"></script>
+  <script src="https://cdn.jsdelivr.net/npm/reveal.js@4.6.1/plugin/search/search.min.js" crossorigin="anonymous"></script>
+  <script src="https://cdn.jsdelivr.net/npm/reveal.js@4.6.1/plugin/math/math.min.js" crossorigin="anonymous"></script>
+  <script src="https://cdn.jsdelivr.net/npm/reveal.js@4.6.1/plugin/zoom/zoom.min.js" crossorigin="anonymous"></script>
+
+  
+  
+    <script src="https://cdn.jsdelivr.net/npm/reveal.js-menu@2.1.0/plugin.js" integrity="sha256-M6JwAjnRAWmi+sbXURR/yAhWZKYhAw7YXnnLvIxrdGs=" crossorigin="anonymous"></script>
+    <script src="https://cdn.jsdelivr.net/npm/reveal.js-menu@2.1.0/menu.js" integrity="sha256-l14dklFcW5mWar6w/9KaW0fWVerf3mYr7Wt0+rXzFAA=" crossorigin="anonymous"></script>
+    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/reveal.js-menu@2.1.0/menu.css" integrity="sha256-0fU8HKLaTjgzfaV9CgSqbsN8ilA3zo6zK1M6rlgULd8=" crossorigin="anonymous">
+  
+
+  
+  
+
+  
+  
+  <script src="/js/wowchemy-slides.js"></script>
+
+</body>
+</html>
diff --git a/slides/index.xml b/slides/index.xml
new file mode 100644
index 0000000..0c16e63
--- /dev/null
+++ b/slides/index.xml
@@ -0,0 +1,135 @@
+<?xml version="1.0" encoding="utf-8" standalone="yes" ?>
+<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
+  <channel>
+    <title>Slides | FS</title>
+    <link>https://frankschae.github.io/slides/</link>
+      <atom:link href="https://frankschae.github.io/slides/index.xml" rel="self" type="application/rss+xml" />
+    <description>Slides</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Tue, 05 Feb 2019 00:00:00 +0000</lastBuildDate>
+    <item>
+      <title>Slides</title>
+      <link>https://frankschae.github.io/slides/example/</link>
+      <pubDate>Tue, 05 Feb 2019 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/slides/example/</guid>
+      <description>&lt;h1 id=&#34;create-slides-in-markdown-with-hugo-blox-builder&#34;&gt;Create slides in Markdown with Hugo Blox Builder&lt;/h1&gt;
+&lt;p&gt;&lt;a href=&#34;https://hugoblox.com/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Hugo Blox Builder&lt;/a&gt; | &lt;a href=&#34;https://docs.hugoblox.com/content/slides/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Documentation&lt;/a&gt;&lt;/p&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;features&#34;&gt;Features&lt;/h2&gt;
+&lt;ul&gt;
+&lt;li&gt;Efficiently write slides in Markdown&lt;/li&gt;
+&lt;li&gt;3-in-1: Create, Present, and Publish your slides&lt;/li&gt;
+&lt;li&gt;Supports speaker notes&lt;/li&gt;
+&lt;li&gt;Mobile friendly slides&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;controls&#34;&gt;Controls&lt;/h2&gt;
+&lt;ul&gt;
+&lt;li&gt;Next: &lt;code&gt;Right Arrow&lt;/code&gt; or &lt;code&gt;Space&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Previous: &lt;code&gt;Left Arrow&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Start: &lt;code&gt;Home&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Finish: &lt;code&gt;End&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Overview: &lt;code&gt;Esc&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Speaker notes: &lt;code&gt;S&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Fullscreen: &lt;code&gt;F&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;Zoom: &lt;code&gt;Alt + Click&lt;/code&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://revealjs.com/pdf-export/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PDF Export&lt;/a&gt;&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;code-highlighting&#34;&gt;Code Highlighting&lt;/h2&gt;
+&lt;p&gt;Inline code: &lt;code&gt;variable&lt;/code&gt;&lt;/p&gt;
+&lt;p&gt;Code block:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-python&#34; data-lang=&#34;python&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;porridge&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s2&#34;&gt;&amp;#34;blueberry&amp;#34;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;porridge&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;s2&#34;&gt;&amp;#34;blueberry&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;nb&#34;&gt;print&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s2&#34;&gt;&amp;#34;Eating...&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;hr&gt;
+&lt;h2 id=&#34;math&#34;&gt;Math&lt;/h2&gt;
+&lt;p&gt;In-line math: $x + y = z$&lt;/p&gt;
+&lt;p&gt;Block math:&lt;/p&gt;
+&lt;p&gt;$$
+f\left( x \right) = ;\frac{{2\left( {x + 4} \right)\left( {x - 4} \right)}}{{\left( {x + 4} \right)\left( {x + 1} \right)}}
+$$&lt;/p&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;fragments&#34;&gt;Fragments&lt;/h2&gt;
+&lt;p&gt;Make content appear incrementally&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{% fragment %}} One {{% /fragment %}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{% fragment %}} **Two** {{% /fragment %}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{% fragment %}} Three {{% /fragment %}}
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Press &lt;code&gt;Space&lt;/code&gt; to play!&lt;/p&gt;
+&lt;span class=&#34;fragment &#34; &gt;
+  One
+&lt;/span&gt;
+&lt;span class=&#34;fragment &#34; &gt;
+  &lt;strong&gt;Two&lt;/strong&gt;
+&lt;/span&gt;
+&lt;span class=&#34;fragment &#34; &gt;
+  Three
+&lt;/span&gt;
+&lt;hr&gt;
+&lt;p&gt;A fragment can accept two optional parameters:&lt;/p&gt;
+&lt;ul&gt;
+&lt;li&gt;&lt;code&gt;class&lt;/code&gt;: use a custom style (requires definition in custom CSS)&lt;/li&gt;
+&lt;li&gt;&lt;code&gt;weight&lt;/code&gt;: sets the order in which a fragment appears&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;speaker-notes&#34;&gt;Speaker Notes&lt;/h2&gt;
+&lt;p&gt;Add speaker notes to your presentation&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-markdown&#34; data-lang=&#34;markdown&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{% speaker_note %}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;-&lt;/span&gt; Only the speaker can read these notes
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;-&lt;/span&gt; Press &lt;span class=&#34;sb&#34;&gt;`S`&lt;/span&gt; key to view
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  {{% /speaker_note %}}
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Press the &lt;code&gt;S&lt;/code&gt; key to view the speaker notes!&lt;/p&gt;
+&lt;aside class=&#34;notes&#34;&gt;
+  &lt;ul&gt;
+&lt;li&gt;Only the speaker can read these notes&lt;/li&gt;
+&lt;li&gt;Press &lt;code&gt;S&lt;/code&gt; key to view&lt;/li&gt;
+&lt;/ul&gt;
+
+&lt;/aside&gt;
+&lt;hr&gt;
+&lt;h2 id=&#34;themes&#34;&gt;Themes&lt;/h2&gt;
+&lt;ul&gt;
+&lt;li&gt;black: Black background, white text, blue links (default)&lt;/li&gt;
+&lt;li&gt;white: White background, black text, blue links&lt;/li&gt;
+&lt;li&gt;league: Gray background, white text, blue links&lt;/li&gt;
+&lt;li&gt;beige: Beige background, dark text, brown links&lt;/li&gt;
+&lt;li&gt;sky: Blue background, thin dark text, blue links&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+&lt;ul&gt;
+&lt;li&gt;night: Black background, thick white text, orange links&lt;/li&gt;
+&lt;li&gt;serif: Cappuccino background, gray text, brown links&lt;/li&gt;
+&lt;li&gt;simple: White background, black text, blue links&lt;/li&gt;
+&lt;li&gt;solarized: Cream-colored background, dark green text, blue links&lt;/li&gt;
+&lt;/ul&gt;
+&lt;hr&gt;
+
+&lt;section data-noprocess data-shortcode-slide
+  
+      
+      data-background-image=&#34;/media/boards.jpg&#34;
+  &gt;
+
+&lt;h2 id=&#34;custom-slide&#34;&gt;Custom Slide&lt;/h2&gt;
+&lt;p&gt;Customize the slide style and background&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-markdown&#34; data-lang=&#34;markdown&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;slide&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;background-image&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;/media/boards.jpg&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;slide&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;background-color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;#0000FF&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;{{&lt;span class=&#34;p&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;slide&lt;/span&gt; &lt;span class=&#34;na&#34;&gt;class&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;my-style&amp;#34;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;&amp;gt;&lt;/span&gt;}}
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;hr&gt;
+&lt;h2 id=&#34;custom-css-example&#34;&gt;Custom CSS Example&lt;/h2&gt;
+&lt;p&gt;Let&amp;rsquo;s make headers navy colored.&lt;/p&gt;
+&lt;p&gt;Create &lt;code&gt;assets/css/reveal_custom.css&lt;/code&gt; with:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-css&#34; data-lang=&#34;css&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nc&#34;&gt;reveal&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;section&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;h1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nc&#34;&gt;reveal&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;section&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nc&#34;&gt;reveal&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;section&lt;/span&gt; &lt;span class=&#34;nt&#34;&gt;h3&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;kc&#34;&gt;navy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;hr&gt;
+&lt;h1 id=&#34;questions&#34;&gt;Questions?&lt;/h1&gt;
+&lt;p&gt;&lt;a href=&#34;https://discord.gg/z8wNYzb&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Ask&lt;/a&gt;&lt;/p&gt;
+&lt;p&gt;&lt;a href=&#34;https://docs.hugoblox.com/content/slides/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Documentation&lt;/a&gt;&lt;/p&gt;
+</description>
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+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
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+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
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+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/software/" />
+<link rel="canonical" href="https://frankschae.github.io/software/" />
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+  <link rel="manifest" href="/manifest.webmanifest" />
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+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
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+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/software/" />
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+    <link rel="alternate" href="/software/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2023-01-16T00:20:37&#43;01:00">
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+  <title>Projects | FS</title>
 
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-  <title>Projects | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   " data-wc-page-id="f06a2a5c6da30d4cf59dafd2c9419459" >
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         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
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         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
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+      
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     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
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-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
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+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
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+      <div class="d-none d-lg-inline-flex">
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+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
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+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
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+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
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+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
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+          </li>
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+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
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+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
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+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
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+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
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+        <li class="nav-item dropdown theme-dropdown">
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+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
+            </a>
+          </div>
         </li>
+        
 
         
         
 
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+  </nav>
+</header>
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+  <i class="far fa-file-alt pub-icon" aria-hidden="true"></i>
+  <a href="https://github.com/gaurav-arya/StochasticAD.jl" target="_blank" rel="noopener">StochasticAD.jl: Automatic Differentiation of Programs with Discrete Randomness</a>
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+    NeurIPS 2022 paper</a>
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+<div class="view-list view-list-item">
+  <i class="far fa-file-alt pub-icon" aria-hidden="true"></i>
+  <a href="https://github.com/JuliaDiff/AbstractDifferentiation.jl" target="_blank" rel="noopener">AbstractDifferentiation.jl: Backend-Agnostic Differentiable Programming in Julia</a>
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2109.12449" target="_blank" rel="noopener">
+    NeurIPS 2021 workshop</a>
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+  <i class="far fa-file-alt pub-icon" aria-hidden="true"></i>
+  <a href="https://sciml.ai/" target="_blank" rel="noopener">SciML Scientific Machine Learning Software</a>
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-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
-        </li>
+<div class="view-list view-list-item">
+  <i class="far fa-file-alt pub-icon" aria-hidden="true"></i>
+  <a href="https://github.com/mschauer/MitosisStochasticDiffEq.jl" target="_blank" rel="noopener">MitosisStochasticDiffEq.jl</a>
 
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+  
 
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-      <li class="nav-item">
-        <a class="nav-link js-search" href="#"><i class="fas fa-search" aria-hidden="true"></i></a>
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-        <a href="#" class="nav-link js-theme-selector" data-toggle="dropdown" aria-haspopup="true">
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+</div>
   </div>
-</nav>
 
+  <div class="page-footer">
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+    <div class="container">
+      <footer class="site-footer">
 
   
 
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-  <div class="container">
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         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
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index 9251c71..3ff6afa 100644
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+<!DOCTYPE html>
+<html lang="en-us">
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diff --git a/software/sciml/index.html b/software/sciml/index.html
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--- a/software/sciml/index.html
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-<!DOCTYPE html><html lang="en-us" >
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+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
   
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index 3f176ee..71af370 100644
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-<!DOCTYPE html><html lang="en-us" >
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+        Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris&rsquo; article3 for a detailed examination.
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+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sun, 01 Aug 2021 12:03:17 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
       <title>AbstractDifferentiation.jl</title>
       <link>https://frankschae.github.io/tag/abstractdifferentiation.jl/</link>
     </image>
@@ -17,33 +17,17 @@
       <link>https://frankschae.github.io/post/abstract_differentiation/</link>
       <pubDate>Sun, 01 Aug 2021 12:03:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/abstract_differentiation/</guid>
-      <description>&lt;p&gt;
-&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is 
-&lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&#39; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and 
-&lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, 
-&lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different 
-&lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by 
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a 
-&lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
-&lt;p&gt;
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
+      <description>&lt;p&gt;&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is &lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&amp;rsquo; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and &lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, &lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different &lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by &lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a &lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
+&lt;p&gt;&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
 &lt;h2 id=&#34;optimization-of-the-rosenbrock-function&#34;&gt;Optimization of the Rosenbrock function&lt;/h2&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
+&lt;p&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
 &lt;p&gt;$$
 g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2.
 $$&lt;/p&gt;
 &lt;p&gt;The function $g$ has a global minimum at $(x_1^\star, x_2^\star)= (a, a^2)$ with $g(x_1^\star, x_2^\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.&lt;/p&gt;
-&lt;p&gt;Let us now implement the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
+&lt;p&gt;Let us now implement the &lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
 &lt;p&gt;$$
 S({\bf x}) = \frac{1}{2} \sum_{i=1}^M f_i({\bf x})^2.
 $$&lt;/p&gt;
@@ -51,298 +35,275 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 {\bf x}^{k+1} = {\bf x}^k - \alpha_k \left(J^T J \right)^{-1} J^T f({\bf x}^k),
 $$
-where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a 
-&lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
+where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a &lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;The following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\bf x^\star}=(1,1)$ and our initial guess ${\bf x_0}=(0,-0.1)$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Pkg
-path = @__DIR__
-cd(path); Pkg.activate(&amp;quot;.&amp;quot;); Pkg.instantiate()
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@__DIR__&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;activate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;.&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;instantiate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## AbstractDifferentiation is not released yet!!&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AbstractDifferentiation&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Enzyme&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;UnPack&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Rosenbrock function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualization&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt;    
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.6&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.5&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;linealpha&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;colorbar&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;camera&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;heatmap&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_0&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:circle&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:green&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x^\star&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:star&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:red&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Rosenbrock.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-## AbstractDifferentiation is not released yet!!
 
-using AbstractDifferentiation
-using Test, LinearAlgebra
-using FiniteDifferences, ForwardDiff, Zygote
-using Enzyme, UnPack
-using Plots, LaTeXStrings
-# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6
 
-## Rosenbrock function
-# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²
-g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2
 
-# visualization
-p = [1.0,100.0]
-x₀ = [0.0,-0.1]
-xopt = [1.0,1.0]
 
-do_plot = true
-if do_plot    
-    x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5
-    z = Surface((x₁,x₂)-&amp;gt;g([x₁,x₂],p), x₁, x₂)
-    pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true,
-                labelfontsize=20,camera = (3,50),
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
 
-    pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp),
-                labelfontsize=20,
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
-    scatter!(pl2, [(x₀[1],x₀[2])], label=L&amp;quot;x_0&amp;quot;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green)
-    scatter!(pl2, [(xopt[1],xopt[2])],label=L&amp;quot;x^\star&amp;quot;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red)
 
-    pl = plot(pl1,pl2, layout=(2,1))
-    savefig(pl, &amp;quot;Rosenbrock.png&amp;quot;)
-end
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
 
 
-
-
-
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Rosenbrock.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;To apply the Gauss-Newton algorithm to the Rosenbrock function $g$, we first cast $g$ into an appropriate form fulfilling $S({\bf x})$, i.e., we use:&lt;/p&gt;
 &lt;p&gt;$$
 f:\mathbb{R}^2\rightarrow\mathbb{R}^2:  {\bf x} \mapsto \begin{pmatrix}
-f_1({\bf x}) \\&lt;br&gt;
-f_2({\bf x}) \\&lt;br&gt;
+f_1({\bf x}) \\
+f_2({\bf x}) \\
 \end{pmatrix} = \begin{pmatrix}
-\sqrt{2}(a-x_1) \\&lt;br&gt;
-\sqrt{2b}(x_2-x_1^2)\\&lt;br&gt;
+\sqrt{2}(a-x_1) \\
+\sqrt{2b}(x_2-x_1^2)\\
 \end{pmatrix},
 $$&lt;/p&gt;
 &lt;p&gt;instead of $g$. We can easily compute the Jacobian of $f$ manually&lt;/p&gt;
 &lt;p&gt;$$
 J =  \begin{pmatrix}
--\sqrt{2} &amp;amp; 0 \\&lt;br&gt;
--2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\&lt;br&gt;
+-\sqrt{2} &amp;amp; 0 \\
+-2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\
 \end{pmatrix}.
 $$&lt;/p&gt;
 &lt;p&gt;We can then implement a (simple, non-optimized) version of the Gauss-Newton algorithm as follows.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# bring Rosenbrock function into the form &amp;quot;sum of squares of functions&amp;quot;
-f1(x,p) = convert(eltype(x),sqrt(2))*(p[1]-x[1])
-f2(x,p) = convert(eltype(x),sqrt(2*p[2]))*(x[2]-x[1]^2)
-f(x,p) = [f1(x,p),f2(x,p)]
-function f(res,x,p) # Enzyme works with inplace functions
-	res[1] = f1(x,p)
-	res[2] = f2(x,p)
-	return nothing
-end
-
-## manually pre-defined Jacobian
-function Jacobian(x,p)
-  [-convert(eltype(x),sqrt(2))   0
-  -2*x[1]*convert(eltype(x),sqrt(2*p[2]))  convert(eltype(x),sqrt(2*p[2]))]
-end
-
-## Gauss-Newton scheme
-function GaussNewton!(xs, x, p; maxiter=8, backend=nothing)
-    for i=1:maxiter
-        x = step(x, p, backend)
-        @info i
-        @show x
-        push!(xs, x)
-    end
-    return xs, x
-end
-done(x,x2,p) = g(x2,p) &amp;lt; g(x,p)
-function step(x, p, backend::Nothing, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = Jacobian(x,p)
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;xs = [x₀]
-GaussNewton!(xs, x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# output:
-[ Info: 1 ]
-x = [0.125, -0.08750000000000001]
-[ Info: 2 ]
-x = [0.234375, -0.047265625000000006]
-[ Info: 3 ]
-x = [0.4257812499999995, 0.06800537109374968]
-[ Info: 4 ]
-x = [0.5693359374999986, 0.21857223510742047]
-[ Info: 5 ]
-x = [0.784667968749996, 0.5165503501892037]
-[ Info: 6 ]
-x = [0.9999999999999961, 0.9536321163177449]
-[ Info: 7 ]
-x = [0.9999999999999989, 0.9999999999999999]
-[ Info: 8 ]
-x = [1.0, 1.0]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of 
-&lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## FiniteDifferences
-struct FDMBackend{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend() = FDMBackend(central_fdm(5, 1))
-const fdm_backend = FDMBackend()
-# Minimal interface
-AD.@primitive function jacobian(ab::FDMBackend, f, xs...)
-    return FiniteDifferences.jacobian(ab.alg, f, xs...)
-end
-
-# AD Jacobian returns tuple
-# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]
-# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]
-
-@test AD.jacobian(fdm_backend, x-&amp;gt;f(x,p), x₀)[1] ≈ Jacobian(x₀, p)
-@test AD.jacobian(fdm_backend, f, x₀, p)[1] ≈ Jacobian(x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function step(x, p, backend, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = AD.jacobian(backend, f, x, p)[1]
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-
-
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=fdm_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If we want to use reverse-mode AD instead, for example via 
-&lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Zygote
-struct ZygoteBackend &amp;lt;: AD.AbstractReverseMode end
-const zygote_backend = ZygoteBackend()
-AD.@primitive function pullback_function(ab::ZygoteBackend, f, xs...)
-    return function (vs)
-        # Supports only single output
-        _, back = Zygote.pullback(f, xs...)
-        if vs isa AbstractVector
-            back(vs)
-        else
-            @assert length(vs) == 1
-            back(vs[1])
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(zygote_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=zygote_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (
-&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## ForwardDiff
-struct ForwardDiffBackend &amp;lt;: AD.AbstractForwardMode end
-const forwarddiff_backend = ForwardDiffBackend()
-AD.@primitive function pushforward_function(ab::ForwardDiffBackend, f, xs...)
-    # jvp = f&#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0
-    return function (vs)
-        if xs isa Tuple
-            @assert length(xs) &amp;lt;= 2
-            if length(xs) == 1
-                (ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1]),0),)
-            else
-                ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1], xs[2]+h*vs[2]),0)
-            end
-        else
-            ForwardDiff.derivative(h-&amp;gt;f(xs+h*vs),0)
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(forwarddiff_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=forwarddiff_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;where we have used that the Jacobian-vector product $f&#39;(x)v$, i.e., the primitives of forward-mode AD, can be computed by 
-&lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. 
-&lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Enzyme
-struct EnzymeBackend{T1,T2,T3,T4} &amp;lt;: AD.AbstractReverseMode
-    out::T1
-    λ::T2
-    ∂f_∂x::T3
-    ∂f_∂p::T4
-end
-
-out = zero(x₀)
-λ = zero(x₀)
-∂f_∂x = zero(x₀)
-∂f_∂p = zero(p)
-
-const enzyme_backend = EnzymeBackend(out,λ,∂f_∂x,∂f_∂p)
-AD.@primitive function pullback_function(ab::EnzymeBackend, f, xs...)
-    return function (vs)  
-        # enzyme works only with inplace functions
-        if !(vs isa AbstractVector)
-            @assert length(vs) == 1 # Supports only single output
-            vs = vs[1]
-        end
-
-        if xs isa Tuple
-            @assert length(xs) == 2  # hard-coded for use case with two inputs
-            x₀ = xs[1]
-            p = xs[2]
-        end
-
-        @unpack out, λ, ∂f_∂x, ∂f_∂p = ab # cached in the struct, could also be created in here
-
-        ∂f_∂x .*= false
-        ∂f_∂p .*= false
-        out .*= false
-
-        copyto!(λ, vs)
-
-        autodiff(Duplicated(out, λ), Duplicated(x₀, ∂f_∂x), Duplicated(p, ∂f_∂p)) do _out,_x, _p
-            f(_out,_x,_p)
-        end
-        return (∂f_∂x,∂f_∂p)
-    end
-end
-AD.isinplace(ab::EnzymeBackend) = true
-AD.primalvalue(ab::EnzymeBackend, nothing, f, xs) = (f(ab.out,xs...);return ab.out)
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(enzyme_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=enzyme_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# bring Rosenbrock function into the form &amp;#34;sum of squares of functions&amp;#34;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Enzyme works with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## manually pre-defined Jacobian&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;   &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Gauss-Newton scheme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# output:&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.125&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.08750000000000001&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.234375&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.047265625000000006&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.4257812499999995&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.06800537109374968&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5693359374999986&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.21857223510742047&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.784667968749996&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5165503501892037&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;6&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999961&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9536321163177449&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;7&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999989&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9999999999999999&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of &lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## FiniteDifferences&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Minimal interface&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# AD Jacobian returns tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If we want to use reverse-mode AD instead, for example via &lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## ForwardDiff&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractForwardMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c&#34;&gt;# jvp = f&amp;#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;where we have used that the Jacobian-vector product $f&amp;rsquo;(x)v$, i.e., the primitives of forward-mode AD, can be computed by &lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. &lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Enzyme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;  
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# enzyme works only with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;  &lt;span class=&#34;c&#34;&gt;# hard-coded for use case with two inputs&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@unpack&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# cached in the struct, could also be created in here&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;autodiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;do&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;isinplace&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;primalvalue&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
 &lt;h2 id=&#34;some-current-glitches&#34;&gt;Some current glitches&lt;/h2&gt;
 &lt;p&gt;First, the push forward of a tuple of vectors, e.g., $(v_1, v_2)$, for a function with several input arguments is currently ambiguous. While &lt;code&gt;AD.jacobian&lt;/code&gt; primitives and &lt;code&gt;AD.pullback_function&lt;/code&gt; primitives interpret the push forward of our $f$ function as&lt;/p&gt;
 &lt;p&gt;$$
@@ -352,86 +313,77 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\partial f(x_0,p)}{\partial x} v_1 + \frac{\partial f(x_0,p)}{\partial p} v_2.
 $$&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous
-vs = (randn(2), randn(2))
-res1 = AD.pushforward_function(fdm_backend, f, x₀, p)(vs)
-res2 = AD.pushforward_function(forwarddiff_backend, f, x₀, p)(vs)
-
-@test res2 ≈ res1[1] + res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;res2a = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((vs[1], zero(vs[2])))
-res2b = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((zero(vs[1]), vs[2]))
-
-@test res2a ≈ res1[1]
-@test res2b ≈ res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;struct FDMBackend2{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend2() = FDMBackend2(central_fdm(5, 1))
-const fdm_backend2 = FDMBackend2()
-AD.@primitive function pushforward_function(ab::FDMBackend2, f, xs...)
-    return function (vs)
-        FDM.jvp(ab.alg, f, tuple.(xs, vs)...)
-    end
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian only defined with respect to single input variable
-@test_throws AssertionError H1 = AD.hessian(forwarddiff_backend, g, x₀, p)
-H1 = AD.hessian(forwarddiff_backend, x-&amp;gt;g(x,p), x₀)
-H2 = AD.hessian(forwarddiff_backend, p-&amp;gt;g(x₀,p), p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)
-backends = AD.HigherOrderBackend((forwarddiff_backend,zygote_backend))
-H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;FDM&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jvp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tuple&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian only defined with respect to single input variable&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test_throws&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AssertionError&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;HigherOrderBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;There are many other use cases, e.g.,&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;AbstractDifferentiation.jl is by no means complete yet. We are still in the very early stages, but we hope to make significant progress in the coming weeks. Some of the next steps are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;fixing remaining bugs, e.g., with respect to the computation of the Hessian and&lt;/li&gt;
-&lt;li&gt;adding AD/Finite Differentiation packages such as 
-&lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
+&lt;li&gt;adding AD/Finite Differentiation packages such as &lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
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diff --git a/tag/adjoint-sensitivity-methods/index.html b/tag/adjoint-sensitivity-methods/index.html
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+<div class="universal-wrapper pt-3">
+  <h1>Adjoint Sensitivity Methods</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  
+
+
+  
+
+  
+  
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+  
+
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a>
+    </div>
+
+    
+    <a href="/post/bouncing_ball/"  class="summary-link">
+      <div class="article-style">
+        In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
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+      
         
 
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-        
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+<div class="article-metadata">
+
+  
+  
+  
+  
+  <div>
+    
+
+  <span >
+      Frank Schäfer and Moritz Schauer</span>
+  </div>
+  
+  
+
+  
+  <span class="article-date">
+    
+    
+      
+    
+    Jul 16, 2021
+  </span>
+  
+
+  
+
+  
+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    19 min read
+  </span>
+  
+
+  
+  
+  
+  
+
+  
+  
+
+</div>
+
+      
+    </div>
+
+    
+
+  </div>
+  <div class="ml-3">
+    
+    
+  </div>
+</div>
+
+  
+    
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+
+  
+
+
+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
+    </div>
+
+    
+    <a href="/post/shadowing/"  class="summary-link">
+      <div class="article-style">
+        In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
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+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/hybridde/" >Neural Hybrid Differential Equations</a>
+    </div>
+
+    
+    <a href="/post/hybridde/"  class="summary-link">
+      <div class="article-style">
+        I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
+      </div>
+    </a>
+    
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+      <a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a>
+    </div>
+
+    
+    <a href="/project/dp_for_control/"  class="summary-link">
+      <div class="article-style">
+        Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
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+  <span >
+      FS in Collaboration With Pavel Sekatski</span>, <span >
+      Martin Koppenhöfer</span>, <span >
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=4" target="_blank" rel="noopener">
+    SIAM CSE 2021 talk</a>
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+    CMD2020GEFES talk</a>
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+    
+    
+      
+      
+      <a href="/project/dp_for_control/" >
+        <img src="/project/dp_for_control/featured_hudf85e1c26ad74a07a00aa9287e088b01_236898_6e426bbd03dcc15fb72201019eec9119.webp" height="113" width="150"
+            alt="Control of (Stochastic) Quantum Dynamics with Differentiable Programming" loading="lazy">
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+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a>
+    </div>
+
+    
+    <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/"  class="summary-link">
+      <div class="article-style">
+        First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
+      </div>
+    </a>
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-      <h2><a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a></h2>
-      <div class="article-style">
-        
-          In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
-        
-      </div>
-    </div>
   
-    
-    
-    
-    <div>
-      <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></h2>
-      <div class="article-style">
-        
-          In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
-        
-      </div>
-    </div>
+
   
-    
-    
-    
-    <div>
-      <h2><a href="/post/hybridde/" >Neural Hybrid Differential Equations</a></h2>
-      <div class="article-style">
-        
-          I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
-        
-      </div>
-    </div>
   
     
-    
-    
-    <div>
-      <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a></h2>
-      <div class="article-style">
-        
-          Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
-        
-      </div>
-    </div>
+  
   
     
-    
-    
-    <div>
-      <h2><a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a></h2>
-      <div class="article-style">
-        
-          First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
-        
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-    </div>
   
 
   
 
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diff --git a/tag/adjoint-sensitivity-methods/index.xml b/tag/adjoint-sensitivity-methods/index.xml
index 56e9430..e690c71 100644
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+    <description>Adjoint Sensitivity Methods</description>
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-      <title>Adjoint sensitivity methods</title>
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+      <title>Adjoint Sensitivity Methods</title>
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     </image>
     
@@ -20,96 +20,106 @@
       <description>&lt;p&gt;In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. We highlight differences between explicit&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and implicit events&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;. As a paradigmatic example, we consider a bouncing ball described by the ODE&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\&lt;br&gt;
+\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\
 \text{d}v(t) &amp;amp;= -\mathrm g\thinspace  \text{d}t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;with initial condition&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t=0) &amp;amp;= z_0 = 5, \\&lt;br&gt;
+z(t=0) &amp;amp;= z_0 = 5, \\
 v(t=0) &amp;amp;= v_0 = -0.1.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;The initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.&lt;/p&gt;
-&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
+&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
 &lt;p&gt;When ignoring the bounces, we can straightforwardly integrate the ODE analytically&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\&lt;br&gt;
+z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\
 v(t) &amp;amp;= v_0 - \mathrm g\thinspace  t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;or numerically using the OrdinaryDiffEq package from the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### simulate forward
+&lt;p&gt;or numerically using the OrdinaryDiffEq package from the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### simulate forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# dynamics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# parameters and solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;5.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.9&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward_no_bounce.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
-using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity
-using Plots, LaTeXStrings
 
-# dynamics
-function f(du,u,p,t)
-  du[1] = u[2]
-  du[2] = -p[1]
-end
 
-# parameters and solve
-z0 = 5.0
-v0 = -0.1
-t0 = 0.0
-tend = 1.9
-g = 10
-γ = 0.8
 
-u0 = [z0,v0]
-tspan = (t0,tend)
-p = [g, γ]
-prob = ODEProblem(f,u0,tspan,p)
 
-# plot forward trajectory
-sol = solve(prob,Tsit5(),saveat=0.1)
-pl = plot(sol, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomleft)
-hline!(pl, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34;&gt;&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
+&lt;/p&gt;
 
 
 
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+
+
+
+
+
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward_no_bounce.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Of course, this way the ball continues to fall through the barrier at $z=0$.&lt;/p&gt;
 &lt;h2 id=&#34;forward-simulation-with-events&#34;&gt;Forward simulation with events&lt;/h2&gt;
 &lt;p&gt;At time $\tau$ around $\tau \approx 1$, the ball hits the ground $z(\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\tau$ with new initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau}) &amp;amp;=  z(\tau-) ,\\&lt;br&gt;
+z({\tau}) &amp;amp;=  z(\tau-) ,\\
 v({\tau})&amp;amp;= -\gamma v(\tau-) ,
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -126,15 +136,15 @@ $$&lt;/p&gt;
 &lt;p&gt;The full explicit trajectory $z_{\rm exp}(t) = z(t)$ is determined by&lt;/p&gt;
 &lt;p&gt;$$
 z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\&lt;br&gt;
--0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\
+-0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\
 \quad  - (-0.99005 + t) (-0.99005 \mathrm g + v_0)\gamma  ,&amp;amp; \forall t \ge \tau,
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;where we used&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\&lt;br&gt;
+z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\
 v({\tau})&amp;amp;= -\gamma v({\tau-}) = -\gamma(v_0 - 0.99005 \mathrm g)  .
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -143,15 +153,14 @@ $$&lt;/p&gt;
 a(z,v) = (z, -\gamma v).
 $$&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;DiscreteCallback&lt;/code&gt; in this case to simulate the system.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with DiscreteCallback (explicit event)
-tstar = (v0 + sqrt(v0^2+2*z0*g))/g
-condition1(u,t,integrator) = (t == tstar)
-affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2]
-cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true))
-
-sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar])
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with DiscreteCallback (explicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiscreteCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstops&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
 &lt;h3 id=&#34;implicit-events&#34;&gt;Implicit events&lt;/h3&gt;
 &lt;p&gt;The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)&lt;/p&gt;
 &lt;p&gt;$$
@@ -160,7 +169,7 @@ $$&lt;/p&gt;
 &lt;p&gt;where an event occurs at time $\tau$ if $g(z(\tau),v(\tau),p,\tau) = 0$. We have already used this condition to define our impact time $\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time  $g(z,v,p,t) = 0$.&lt;/p&gt;
 &lt;p&gt;As in the previous case, we can analytically compute the full trajectory of the ball. By substituting the formula for $\tau$ we have at the event time&lt;/p&gt;
 &lt;p&gt;\begin{aligned}
-z({\tau})&amp;amp;= 0, \\&lt;br&gt;
+z({\tau})&amp;amp;= 0, \\
 v({\tau}-)&amp;amp;= - \sqrt{v_0^2 + 2 \mathrm g z_0}
 \end{aligned}&lt;/p&gt;
 &lt;p&gt;for the left limit and&lt;/p&gt;
@@ -170,33 +179,27 @@ v({\tau})&amp;amp;= \gamma \sqrt{v_0^2 + 2 \mathrm g z_0}
 &lt;p&gt;right after the bounce. Thus, the full trajectory $z_{\rm imp}(t) = z(t)$ is given by&lt;/p&gt;
 &lt;p&gt;$$
 (\star) \quad z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\&lt;br&gt;
--\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\
+-\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\
 \quad\cdot \space (-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0} (1 + 2 \gamma)), &amp;amp; \forall t \ge \tau.
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;This is correct even if one substitutes, e.g., a value with higher precision $\mathrm g = 9.81$ for the gravitation constant.&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;ContinuousCallback&lt;/code&gt; in this case.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with ContinuousCallback (implicit event)
-condition2(u,t,integrator) = u[1] # Event happens when condition2(u,t,integrator) == 0
-cb2 = ContinuousCallback(condition2,affect!,save_positions=(true,true))
-sol2 = solve(prob,Tsit5(),callback=cb2,saveat=0.1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# plot forward trajectory
-pl1 = plot(sol1, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;explicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-pl2 = plot(sol2, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;implicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-hline!(pl1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-hline!(pl2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-pl = plot(pl1,pl2)
-savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with ContinuousCallback (implicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Event happens when condition2(u,t,integrator) == 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ContinuousCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;explicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;implicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-  
 
 
 
@@ -208,18 +211,15 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In addition, the implicitly defined impact time via the &lt;code&gt;ContinuousCallback&lt;/code&gt; also changes appropriately when changing the initial conditions or the parameters, for example when using  $\mathrm g = 9.81$ for the gravitation constant. In other words, the event time $\tau=\tau(p,z_0,v_0,t_0)$ is a function of the parameters and initial conditions, and is implicitly defined by the event condition.&lt;/p&gt;
 &lt;p&gt;Suppose we let the ball drop from a somewhat higher position now. Does an increase in height $z$ at $t=0$ give an increase or decrease in height at the end time $t_\text{end}=1.9$? This is something we can answer with sensitivity analysis. For example if we increase the height by (a fraction of) one unit then using $(\star)$&lt;/p&gt;
@@ -228,36 +228,34 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 $$&lt;/p&gt;
 &lt;p&gt;meaning the height at $t_\text{end}$ is also by a corresponding fraction of 0.84 units higher.&lt;/p&gt;
 &lt;p&gt;We can verify this visually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# animate forward trajectory
-sol3 = solve(remake(prob,u0=[u0[1]+0.5,u0[2]]),Tsit5(),callback=cb2,saveat=0.01)
-
-plt2 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;, xlims=(t0,tend))
-hline!(plt2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-plot!(plt2, sol3, tspan=(t0,tend), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)
-
-list_plots = []
-for t in sol3.t
-  tstart = 0.0
-
-  plt1 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;)
-  hline!(plt1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-  plot!(plt1, sol3, tspan=(t0,t), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-  scatter!(plt1,[t,t], sol3(t), color=[1, 2], label=false)
-  plt = plot(plt1,plt2)
-  push!(list_plots, plt)
-end
-
-plot(list_plots[100])
-
-anim = animate(list_plots,every=1)
-&lt;/code&gt;&lt;/pre&gt;
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# animate forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]]),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
-  
 
 
 
@@ -269,18 +267,12 @@ anim = animate(list_plots,every=1)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/BB.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;The original curve is shown in black in the figure above.&lt;/p&gt;
 &lt;h2 id=&#34;sensitivity-analysis-with-events&#34;&gt;Sensitivity analysis with events&lt;/h2&gt;
@@ -295,7 +287,7 @@ $$&lt;/p&gt;
 &lt;p&gt;Inserting our results for $z(t) = z_{\rm exp}(t)$ instead of $z(t) = z_{\rm imp}(t)$ at $t_{\text{end}}$, a different value for the sensitivity is obtained (a value ignoring the changes in $z$ due to changes in the bouncing time $\tau$), e.g.,&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L^{\text{exp}}}{\text{d} z_0} = \begin{cases}
-1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\&lt;br&gt;
+1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\
 0.84 &amp;amp; \text{with } z(t) = z_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -306,7 +298,7 @@ $$&lt;/p&gt;
 &lt;p&gt;we could be interested in the sensitivity of $L$ with respect $g$. Using the velocity $v(t)$ at the explicit time $t = 0.9905$ instead of the implicit $v(\tau-)$, again a different value for the sensitivity is obtained (a value again ignoring the changes in $\tau$ and not the one we are looking for), even though $\tau = 0.9905$ in both cases:&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L}{\text{d} g} = \begin{cases}
--0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\&lt;br&gt;
+-0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\
 -\frac{g}{\sqrt{v_0^2 + 2 g z_0}} = -0.99995 &amp;amp; \text{ with } v(t) = v_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -323,51 +315,48 @@ $$&lt;/p&gt;
 &lt;p&gt;We write&lt;/p&gt;
 &lt;p&gt;$$\text{solve}(t_0, x_0, t, p)$$&lt;/p&gt;
 &lt;p&gt;$(= x(t))$ for the functional solution of the ODE at time $t$.&lt;/p&gt;
-&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
+&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
 &lt;p&gt;Let us focus on the &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; algorithm which computes the sensitivities&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\&lt;br&gt;
+\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\
 \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}p} &amp;amp;= \lambda_{p}(t_{0}),
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;of a loss function $L$ acting on the final state with respect to the initial state and the parameters. It does so by solving an ODE for $\lambda(s)$ in reverse time from $t$ to $t_0$&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\&lt;br&gt;
+\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\
 \frac{\text{d}\lambda_{p}(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(x(s), \rightarrow p, s)}{\text{d} p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;with initial conditions:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\&lt;br&gt;
+\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\
 \lambda_{p}(t) &amp;amp;= 0.
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The arrows ($\rightarrow$) indicate the variable with respect to which we differentiate, which will become important later when the same variable shows up in multiple function arguments.&lt;/p&gt;
-&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in &lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;&lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, essentially the custom primitive differentiation rule of &lt;code&gt;solve&lt;/code&gt;, is the elementary building block needed to derive sensitivities also in more complicated examples:&lt;/p&gt;
 &lt;p&gt;Consider a loss depending on the state $x(s_i)$ at fix time points $s_i$ through loss functions $L_i^{\text{exp}}$,&lt;/p&gt;
 &lt;p&gt;$$
 L^{\text{exp}} = \sum_i L_i^{\text{exp}}(s_i, x(s_i), p).
 $$&lt;/p&gt;
 &lt;p&gt;Without contortions we can obtain the sensitivity of $L^{\text{exp}}$ in $p$ (or in $x_0$) using the tool we have. For those a bit familiar with automatic differentiation, this is perhaps easiest to see if we write $L^{\text{exp}}$ as pseudo code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1 = solve(t0, x0, s1, p)
-    L = L1(s1, x1, p)
-    x2 = solve(s1, x1, s2, p)
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\&lt;br&gt;
+\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\
 \frac{\text{d}\lambda_{p}(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(x(t), \rightarrow p, t)}{\text{d} p} - \frac{\text{d} L_i^{\text{exp}}(  x(t),\rightarrow p)}{\text{d} p}^\dagger \delta(t-s_i).
 \end{aligned}
 $$&lt;/p&gt;
@@ -381,24 +370,23 @@ $$&lt;/p&gt;
 $$&lt;/p&gt;
 &lt;p&gt;Now we can even properly define the &lt;code&gt;rrule&lt;/code&gt; of &lt;code&gt;solve&lt;/code&gt; in the sense of &lt;code&gt;DiffRules.jl&lt;/code&gt;.&lt;/p&gt;
 &lt;h3 id=&#34;explicit-events-1&#34;&gt;Explicit events&lt;/h3&gt;
-&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1- = solve(t0, x0, s1, p)
-    L = L1-(s1, x1-, p) # saved before affect
-    x1 = a(x1-, p, s1) # affect
-    L += L1(s1, x1, p) # saved after affect
-    x2- = solve(s1, x1, s2, p)
-    L += L2-(s2, x2-, p)
-    x2 = a(x2-, p, s2) # affect
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
+&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref2:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved before affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved after affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\&lt;br&gt;
+\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\
 \lambda_p({s_j}-) &amp;amp;= \lambda_p({s_j}) -  \lambda({s_j})^\dagger \frac{\text{d} a(x({s_j}-), \rightarrow p, {s_j}-)}{\text{d} p}
 \end{aligned}
 $$&lt;/p&gt;
@@ -412,27 +400,26 @@ $$&lt;/p&gt;
 &lt;p&gt;to put emphasis on this, or equivalently, compute for a unspecified function $L(t, x, p)$ the result of $\frac{\text{d}L}{\text{d}p}$ with&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\&lt;br&gt;
+\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\
 &amp;amp;= \frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;which indicates that changing $p$ influences $L$ both through changes in $\tau_1$ as well as changes in&lt;/p&gt;
 &lt;p&gt;$$\xi_1 = x(\tau_1-).$$&lt;/p&gt;
 &lt;p&gt;This case where we have a loss function $L = L_1$ depending on $\tau_1$, $x(\tau_1)$, and $p$
-was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;p&gt;Therefore, the sensitivity of the event time with respect to parameters $\frac{\text{d}\tau}{\text{d}p}$ must be taken into account.
-Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the 
-&lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
+Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the &lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\&lt;br&gt;
+\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The total derivative&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; inside the bracket is:
 $$
 \begin{aligned}
-\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\&lt;br&gt;
+\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;Since&lt;/p&gt;
@@ -455,8 +442,8 @@ for the second term of $\dfrac{\text{d}\tau_1(p)}{\text{d}p}$.&lt;/p&gt;
 &lt;p&gt;We can now write the gradient as:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\&lt;br&gt;
-+&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\&lt;br&gt;
+\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\
++&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\
 +&amp;amp; \frac{\text{d}L(\rightarrow \tau_1(p), \text{solve}(t_0, x_0, \rightarrow \tau_1(p),  p), p)}{\text{d}\tau_1} \frac{\text{d} \tau_1(p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
@@ -525,7 +512,7 @@ $$&lt;/p&gt;
 &lt;li&gt;If both limits contribute to the loss function, the contributions ${\color{red}\lambda_1}$ and ${\color{blue}\lambda_{a,1}}$ are added.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h4 id=&#34;generalization-several-events&#34;&gt;Generalization: several events&lt;/h4&gt;
-&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
+&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
 &lt;p&gt;$$
 \text{solve}(t_0, x_0, t, p)  = \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p),
 $$&lt;/p&gt;
@@ -533,16 +520,15 @@ $$&lt;/p&gt;
 &lt;p&gt;(A good exercise to get familiar with these type of arguments is to verify&lt;/p&gt;
 &lt;p&gt;$$\frac{\text{d}}{\text{d}s} \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p) = 0.)$$&lt;/p&gt;
 &lt;p&gt;Essentially, we know how to address several events already by considering a loss function&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss2(t0, x0, p)
-    tau1, xi1 = solve2(t0, x0, g, p)
-    L = L1(tau1, xi1, p)
-    tau2, xi2 = solve2(tau1, xi1, g, p)
-    L += L2(tau2, xi2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
 &lt;p&gt;Differentiating by hand, we have the following modification of the method from the previous section, which can be derived choosing a particular function $L$ in the previous section which incorporates the &lt;code&gt;solve&lt;/code&gt; from $\tau_1$ to $t_{\text{end}}$. Note&lt;/p&gt;
 &lt;p&gt;$$
 \lambda_{\tau_1}^0 = \frac{\text{d}(\text{solve}(\rightarrow \tau_1,  a(\xi_1,p), t_{\text{end}}, p))}{\text{d} \tau_1} = - f(a(\xi_1,p), p, \tau_1).
@@ -574,43 +560,38 @@ $$&lt;/p&gt;
 &lt;li&gt;Backpropagate $\lambda_J = {\color{red}\lambda_J} + {\color{blue}\lambda_{a,J}} + {\color{green}\lambda_{c,J}}$ to the next event time $\tau_{J-1}$ and iterate over the remaining $J-1$ events.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
-&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
-&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (
-&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$. &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$.&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the &lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
 $$
 \begin{aligned}
-\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\&lt;br&gt;
-&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\&lt;br&gt;
+\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\
+&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\
 &amp;amp;+ \frac{\text{d}f(t, x_1(t),x_2(\rightarrow t),\dots,x_N(t))}{\text{d}t} + \dots +  \frac{\text{d}f(t, x_1(t),x_2(t),\dots,x_N(\rightarrow t))}{\text{d}t}.
 \end{aligned}
-$$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+$$&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -620,8 +601,7 @@ $$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-
       <pubDate>Fri, 02 Jul 2021 11:08:22 +0200</pubDate>
       <guid>https://frankschae.github.io/post/shadowing/</guid>
       <description>&lt;p&gt;In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &amp;ldquo;butterfly effect&amp;rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.&lt;/p&gt;
-&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
+&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:&lt;/p&gt;
 &lt;p&gt;$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -635,57 +615,56 @@ Under the assumption of ergodicity, $\langle g \rangle_∞$ only depends on $p$.
 &lt;p&gt;One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{pmatrix}
-\text{d}x \\&lt;br&gt;
-\text{d}y \\&lt;br&gt;
-\text{d}z \\&lt;br&gt;
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\&lt;br&gt;
-x(\rho-z) - y\\&lt;br&gt;
-x y - \beta z \\&lt;br&gt;
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
-
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
-
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &amp;quot;initial&amp;quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&amp;quot;attractor&amp;quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Statistics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;QuadGK&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SparseArrays&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# simulate 1 trajectory of the Lorenz system forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;28.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;initial&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;attractor&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -697,79 +676,73 @@ savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.&lt;/p&gt;
-&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
+&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
 &lt;p&gt;$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function compute_objective(sol)
-  quadgk(t-&amp;gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&amp;quot;z(t)&amp;quot;, xlabel=L&amp;quot;t&amp;quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&amp;quot;\langle z\rangle&amp;quot;, lw = 2)
-savefig(pl2, &amp;quot;zsingle.png&amp;quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;quadgk&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;atol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;mean_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zsingle.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# for each value of the parameter, solve 20 times the initial value problem&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# wrap the procedure inside a function depending on p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ribbon&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;ho&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zvsrho.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;margin&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;600&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;300&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;z.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We obtain:&lt;/p&gt;
 
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
 
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&amp;quot;\langle z\rangle&amp;quot;, xlabel=L&amp;quot;\rho&amp;quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &amp;quot;zvsrho.png&amp;quot;)
 
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &amp;quot;z.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We obtain:&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -780,41 +753,30 @@ savefig(pl4, &amp;quot;z.png&amp;quot;)
 
 
 
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/z.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/z.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;z.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$&lt;/p&gt;
 &lt;h2 id=&#34;conventional-forward-mode-sensitivity-analysis-and-finite-differencing&#34;&gt;Conventional forward-mode sensitivity analysis and finite-differencing&lt;/h2&gt;
-&lt;p&gt;For non-chaotic systems, we would just use the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we find diverging values:&lt;/p&gt;
+&lt;p&gt;For non-chaotic systems, we would just use the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_forward&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_calculus&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we find diverging values:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\&lt;br&gt;
+&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$&lt;/p&gt;
@@ -822,44 +784,38 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$&lt;/p&gt;
-&lt;p&gt;Similarly, using 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-     xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&amp;quot;Float32&amp;quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
-
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&amp;quot;Float32&amp;quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-
-savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
+&lt;p&gt;Similarly, using &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;500&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_Floats.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-  
 
 
 
@@ -871,18 +827,15 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Without animation:&lt;/p&gt;
 
@@ -890,9 +843,6 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-  
-
-
 
 
 
@@ -902,60 +852,52 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_Floats.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;Luckily, the 
-&lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
+&lt;p&gt;Luckily, the &lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
 &lt;blockquote&gt;
 &lt;p&gt;Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&amp;ldquo;shadows&amp;rdquo;) the numerically computed one.&lt;/p&gt;
 &lt;/blockquote&gt;
 &lt;h2 id=&#34;shadowing-methods&#34;&gt;Shadowing methods&lt;/h2&gt;
-&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref2:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref3:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;lss-and-nilss-for-the-lorenz-system&#34;&gt;LSS and NILSS for the Lorenz system&lt;/h2&gt;
-&lt;p&gt;Switching to LSS or NILSS within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
+&lt;p&gt;Switching to LSS or NILSS within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# objective&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# LSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;CosWindowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0095888187322035&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Cos2Windowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0343951385924328&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0284286902740765&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_adjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.028428690274077&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# select via sensealg in solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9694728321500617&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;&lt;code&gt;CosWindowing()&lt;/code&gt; (default)&lt;/li&gt;
 &lt;li&gt;&lt;code&gt;Cos2Windowing()&lt;/code&gt;&lt;/li&gt;
@@ -964,72 +906,68 @@ dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
 &lt;p&gt;Additionally, an adjoint implementation &lt;code&gt;AdjointLSS()&lt;/code&gt; is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that &lt;code&gt;AdjointLSS(alpha=10.0)&lt;/code&gt; agrees perfectly with &lt;code&gt;ForwardLSS(alpha=10.0)&lt;/code&gt;. In all cases considered, we find the expected sensitivity value of $\approx 1$.&lt;/p&gt;
 &lt;p&gt;However, the use of &lt;code&gt;LSS()&lt;/code&gt; is (typically) much more expensive than the use of &lt;code&gt;NILSS()&lt;/code&gt;, because &lt;code&gt;LSS()&lt;/code&gt; needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of &lt;code&gt;NILSS()&lt;/code&gt; scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making &lt;code&gt;NILSS()&lt;/code&gt; more efficient.&lt;/p&gt;
 &lt;p&gt;In the &lt;code&gt;NILSS()&lt;/code&gt; algorithm, the user can control the number of steps per segment as well as the number of segments.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# NILSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# make sure trajectory is fully on the attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;120.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9966924374966089&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0416028730638789&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Warning: Detected a large value of ξ at the beginning of a segment.&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref4:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;NILSAS&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;li&gt;FD-NILSS&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref4:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019) &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019)&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019) &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019)&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1038,21 +976,11 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
       <link>https://frankschae.github.io/post/hybridde/</link>
       <pubDate>Wed, 16 Jun 2021 14:50:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/hybridde/</guid>
-      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
+      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
 &lt;p&gt;This first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).&lt;/p&gt;
-&lt;p&gt;Note: Please check the 
-&lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
+&lt;p&gt;Note: Please check the &lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
 &lt;h2 id=&#34;hybrid-differential-equations&#34;&gt;Hybrid Differential Equations&lt;/h2&gt;
-&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
+&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
 &lt;p&gt;Some relevant examples for hybrid DEs with discrete or continuous callbacks are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; (&lt;code&gt;ContinuousCallback&lt;/code&gt;).&lt;/li&gt;
@@ -1062,7 +990,7 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
 &lt;li&gt;pharmacokinetic models&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, where explicit dosing times change the drug concentration in the blood (&lt;code&gt;DiscreteCallback&lt;/code&gt;). The simplest possible example being the one-compartment model.&lt;/li&gt;
 &lt;li&gt;kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (&lt;code&gt;DiscreteCallback&lt;/code&gt;).&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;kicked-harmonic-oscillator&#34;&gt;Kicked harmonic oscillator&lt;/h2&gt;
 &lt;p&gt;Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form&lt;/p&gt;
 &lt;p&gt;$$
@@ -1071,40 +999,33 @@ $$&lt;/p&gt;
 &lt;p&gt;where $a=0.1$ and $b=1$ with initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-x(t=0) &amp;amp;= 1  \\&lt;br&gt;
+x(t=0) &amp;amp;= 1  \\
 v(t=0) &amp;amp;= \dot{x}(t=0) = 0.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;This second-order ODE can be 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use 
-&lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity
-using Zygote
-using Random
-u0 = Float32[1.; 0.]
-
-tspan = (0.0f0,50.0f0)
-
-dtsave = 0.5f0
-t = tspan[1]:dtsave:tspan[2]
-
-function oscillator!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  return nothing
-end
-
-prob_data = ODEProblem(oscillator!,u0,tspan)
-
-# ODE without kicks
-pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
+&lt;p&gt;This second-order ODE can be &lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use &lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqFlux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Optim&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# ODE without kicks&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-  
 
 
 
@@ -1116,44 +1037,36 @@ pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &am
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as &lt;code&gt;1&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;kicktimes = tspan[1]:1:tspan[2]
-function kick!(integrator)
-  integrator.u[end] += one(eltype(integrator.u))
-end
-cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false))
-
-sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t)
-t_data = sol_data.t
-ode_data = Array(sol_data)
-
-# visualize data
-pl1 = plot(t_data,ode_data[1,:],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl1,t_data,ode_data[2,:],label=&amp;quot;data v(t)&amp;quot;)
-
-pl2 = plot(t_data[1:20],ode_data[1,1:20],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl2,t_data[1:20],ode_data[2,1:20],label=&amp;quot;data v(t)&amp;quot;)
-pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualize data&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
-  
 
 
 
@@ -1165,63 +1078,56 @@ pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting &lt;code&gt;save_positions=(true,true)&lt;/code&gt;, the kicks would be saved before &lt;strong&gt;and&lt;/strong&gt; after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;, we now aim at learning (potentially) missing parts of the model from these data traces.&lt;/p&gt;
 &lt;h3 id=&#34;high-domain-knowledge&#34;&gt;High domain knowledge&lt;/h3&gt;
 &lt;p&gt;For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the &lt;code&gt;affect!&lt;/code&gt; function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of &lt;code&gt;integrator.u&lt;/code&gt;. If we further set the initial value of that component to &lt;code&gt;1&lt;/code&gt;, then the neural network only has to learn that &lt;code&gt;du[3]&lt;/code&gt; is &lt;code&gt;0&lt;/code&gt;. In other words, the output of the neural network must be &lt;code&gt;0&lt;/code&gt; for all states &lt;code&gt;u&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;Random.seed!(123)
-nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1))
-p_nn1 = initial_params(nn1)
-
-function f1!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3] = nn1(u[1:2], p)[1]
-  return nothing
-end
-
-affect!(integrator) = integrator.u[2] += integrator.u[3]
-cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false))
-z0 = Float32[u0;one(u0[1])]
-prob1 = ODEProblem(f1!,z0,tspan,p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# to visualize the predictions of the trained neural network below
-function visualize(prob,p)
-  _prob = remake(prob,p=p)
-  ode_pred = Array(solve(_prob,Tsit5(),callback=cb,
-                 saveat=dtsave))[1:2,:]
-  pl1 = plot(t_data,ode_pred[1,:],label=&amp;quot;x(t)&amp;quot;)
-  scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=&amp;quot;data x(t)&amp;quot;)
-  pl2 = plot(t_data,ode_pred[2,:],label=&amp;quot;v(t)&amp;quot;)
-  scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=&amp;quot;data v(t)&amp;quot;)
-
-  pl = plot(pl1, pl2, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-  return pl, sum(abs2,ode_data .- ode_pred)
-end
-
-pl = plot(solve(prob1,Tsit5(),saveat=t,
-  callback=cb
-  ),label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;123&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FastChain&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tanh&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial_params&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# to visualize the predictions of the trained neural network below&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                 &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
-  
 
 
 
@@ -1233,76 +1139,66 @@ pl = plot(solve(prob1,Tsit5(),saveat=t,
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; &gt;
 
-
-  &lt;img src=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;untrained_nn.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;which (of course) doesn&amp;rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### loss function
-function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint())
-  _prob = remake(prob,p=p)
-  pred = Array(solve(_prob,Tsit5(),callback=cb,
-               saveat=dtsave,sensealg=sensealg))[1:2,:]
-  sum(abs2,ode_data .- pred)
-end
-
-loss(p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The recently implemented tools are deeply hidden within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., 
-&lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;###################################
-# training loop
-# optimize the parameters for a few epochs with ADAM
-function train(prob, p_nn; sensealg=BacksolveAdjoint())
-  opt = ADAM(0.0003f0)
-  list_plots = []
-  losses = []
-  for epoch in 1:200
-    println(&amp;quot;epoch: $epoch / 200&amp;quot;)
-    _dy, back = Zygote.pullback(p -&amp;gt; loss(p,
-      prob=prob,
-      sensealg=sensealg), p_nn)
-    gs = @time back(one(_dy))[1]
-    push!(losses, _dy)
-    if epoch % 10 == 0
-      # plot every xth epoch
-      pl, test_loss = visualize(prob, p_nn)
-      println(&amp;quot;Loss (epoch: $epoch): $test_loss&amp;quot;)
-      display(pl)
-      push!(list_plots, pl)
-    end
-    Flux.Optimise.update!(opt, p_nn, gs)
-    println(&amp;quot;&amp;quot;)
-  end
-  return losses, list_plots
-end
-
-# plot training loss
-losses, list_plots = train(prob1, p_nn1)
-pl1 = plot(losses, lw = 1.5, xlabel = &amp;quot;epoch&amp;quot;, ylabel=&amp;quot;loss&amp;quot;, legend=false)
-pl2 = list_plots[end]
-pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t,
-   callback=cb
-  ), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-
-pl = plot(pl2,pl3)
-&lt;/code&gt;&lt;/pre&gt;
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### loss function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ReverseDiffAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;               &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The recently implemented tools are deeply hidden within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., &lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;###################################&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# training loop&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# optimize the parameters for a few epochs with ADAM&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ADAM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0003f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;200&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt; / 200&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;%&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;c&#34;&gt;# plot every xth epoch&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_loss&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Loss (epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;): &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$test_loss&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;display&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Optimise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;update!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot training loss&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;epoch&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;loss&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
-  
 
 
 
@@ -1314,98 +1210,86 @@ pl = plot(pl2,pl3)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/trained1.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/trained1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;trained1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We see the expected constant value of &lt;code&gt;u[3]&lt;/code&gt;, indicating a kick to the velocity of &lt;code&gt;+=1&lt;/code&gt;, at the kicking times over the full time interval.&lt;/p&gt;
 &lt;h2 id=&#34;reducing-the-domain-knowledge&#34;&gt;Reducing the domain knowledge&lt;/h2&gt;
-&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
+&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;changing the initial condition of the third component of &lt;code&gt;u&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;using another affect function &lt;code&gt;affect!(integrator) = integrator.u[2] = integrator.u[3]&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;dropping the knowledge that only &lt;code&gt;u[2]&lt;/code&gt; gets a kick by using a neural network with &lt;code&gt;2&lt;/code&gt; outputs (+ a fourth component in the ODE):&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4]
-function f2!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;fitting the parameters $a$ and $b$ simultaneously:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f3!(du,u,p,t)
-  a = p[end-1]
-  b = p[end]
-  nn_weights = p[1:end-2]
-
-  du[1] = u[2]
-  du[2] = -b*u[1] - a*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f3!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;inferring the entire underlying dynamics using a neural network with &lt;code&gt;4&lt;/code&gt; outputs:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f4!(du,u,p,t)
-  Ω = nn3(u[1:2], p)
-
-  du[1] = Ω[1]
-  du[2] = Ω[2]
-  du[3:4] .= Ω[3:4]
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f4!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;etc.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;refine the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;) and&lt;/li&gt;
-&lt;li&gt;support direct usage through the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
+&lt;li&gt;support direct usage through the &lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021) &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021)&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1423,21 +1307,14 @@ end
       <link>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</link>
       <pubDate>Sat, 30 May 2020 15:10:33 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</guid>
-      <description>&lt;p&gt;First and foremost, I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
+      <description>&lt;p&gt;First and foremost, I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
 Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both  &lt;strong&gt;new high weak order solvers&lt;/strong&gt; as well as &lt;strong&gt;adjoint sensitivity methods&lt;/strong&gt; for stochastic differential equations (SDEs).
-The project is embedded within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
-Ultimately, the planned contributions will allow researchers to simulate (or even 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
+The project is embedded within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
+Ultimately, the planned contributions will allow researchers to simulate (or even &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
 Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.&lt;/p&gt;
 &lt;p&gt;This post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.&lt;/p&gt;
 &lt;h2 id=&#34;high-weak-order-solvers&#34;&gt;High Weak Order Solvers&lt;/h2&gt;
-&lt;p&gt;Currently, the 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
+&lt;p&gt;Currently, the &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
 In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.&lt;/p&gt;
 &lt;p&gt;However in many situations, we are only aiming for computing an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt;.
 In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods.
@@ -1445,7 +1322,7 @@ In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&
 As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.&lt;/p&gt;
 &lt;h3 id=&#34;second-order-runge-kutta-methods-for-ito-sdes&#34;&gt;Second order Runge-Kutta methods for Ito SDEs&lt;/h3&gt;
 &lt;p&gt;In the beginning of the community bonding period I finished the implementations of the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and &lt;code&gt;RI1()&lt;/code&gt;&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;. Specifically, only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn (in contrast to &lt;code&gt;m(m+1)/2&lt;/code&gt; from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.&lt;/p&gt;
-&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -1454,47 +1331,46 @@ $$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e7&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;. The code for a single  &lt;code&gt;dt&lt;/code&gt; reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-numtraj = 1e7
-u₀ = [1.0,1.0]
-function f!(du,u,p,t)
-  du[1] = -273//512*u[1]
-  du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-end
-function g!(du,u,p,t)
-  du[1,1] = 1//4*u[1]
-  du[1,2] = 1//16*u[1]
-  du[2,1] = (1-2*sqrt(2))/4*u[1]
-  du[2,2] = 1//10*u[1]+1//16*u[2]
-end
-dt = 1//8
-tspan = (0.0,10.0)
-prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2))
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;();&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
 
-h(z) = z^2
 
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h(sol[end][1]),false)
-        )
-sol = solve(ensemble_prob, DRI1();
-        dt=dt,
-        save_start=false,
-        save_everystep=false,
-        weak_timeseries_errors=false,
-        weak_dense_errors=false,
-        trajectories=numtraj)
 
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
-&lt;pre&gt;&lt;code&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -1505,30 +1381,19 @@ sol = solve(ensemble_prob, DRI1();
 
 
 
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/DRI1.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/DRI1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;DRI1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In the next couple of weeks, my focus will be on&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;adding other high weak order solvers,&lt;/li&gt;
 &lt;li&gt;implementing adaptive time stepping.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;More of our near-term goals are collected in this 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;More of our near-term goals are collected in this &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;adjoint-sensitivity-methods-for-sdes&#34;&gt;Adjoint Sensitivity Methods for SDEs&lt;/h2&gt;
-&lt;p&gt;In 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
+&lt;p&gt;In &lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
 &lt;p&gt;$$
 L = |f(p)-y|^2.
 $$&lt;/p&gt;
@@ -1537,46 +1402,42 @@ One possibility to obtain the gradient information for (stochastic) differential
 While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.&lt;/p&gt;
 &lt;p&gt;Alternatively to the &amp;ldquo;direct&amp;rdquo; AD approaches, the &lt;strong&gt;adjoint sensitivity method&lt;/strong&gt; can be used&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.
 Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).&lt;/p&gt;
-&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
+&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
 &lt;h3 id=&#34;reverse-sde-time-evolution&#34;&gt;Reverse SDE time evolution&lt;/h3&gt;
 &lt;p&gt;The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms.
 As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed.
 In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values.
-After some fixes for the 
-&lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
+After some fixes for the &lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
 &lt;p&gt;As an example, the code below computes first the forward evolution of an SDE&lt;/p&gt;
 &lt;p&gt;$$ dX  =  \alpha X dt + \beta X dW$$&lt;/p&gt;
 &lt;p&gt;with $\alpha=1.01$, $\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;  using StochasticDiffEq, DiffEqNoiseProcess
-
-  α=1.01
-  β=0.87
-
-  dt = 1e-3
-  tspan = (0.0,1.0)
-  u₀=1/2
-
-  tarray =  collect(tspan[1]:dt:tspan[2])
-
-  f!(du,u,p,t) = du .= α*u
-  g!(du,u,p,t) = du .= β*u
-
-
-  prob = SDEProblem(f!,g!,[u₀],tspan)
-  sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false)
-
-  _sol = deepcopy(sol) # to make sure the plot is correct
-  W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W))
-  prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1)
-  sol1 = solve(prob1,EulerHeun(),dt=dt)
-&lt;/code&gt;&lt;/pre&gt;
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# to make sure the plot is correct&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NoiseGrid&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
-  
 
 
 
@@ -1588,18 +1449,12 @@ After some fixes for the
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/animation.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/animation.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;animation.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gradients-of-diagonal-sdes&#34;&gt;Gradients of diagonal SDEs&lt;/h3&gt;
 &lt;p&gt;I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.&lt;/p&gt;
@@ -1613,82 +1468,80 @@ $$&lt;/p&gt;
 &lt;p&gt;acting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\&lt;br&gt;
+\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\
 \frac{d L}{d \beta}  &amp;amp;= 2 \sum_i W(t_i) |X(t_i)|^2
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Test, LinearAlgebra
-using DiffEqSensitivity, StochasticDiffEq
-using Random
-
-seed = 100
-Random.seed!(seed)
-
-u₀ = [0.5]
-tstart = 0.0
-tend = 0.1
-dt = 0.005
-trange = (tstart, tend)
-t = tstart:dt:tend
-tarray = collect(t)
-
-function g(u,p,t)
-  sum(u.^2.0)
-end
-
-function dg!(out,u,p,t,i)
-  (out.=-2.0*u)
-end
-
-p2 = [1.01,0.87]
-
-f(u,p,t) = p[1]*u
-σ(u,p,t) = p[2]*u
-
-
-Random.seed!(seed)
-prob = SDEProblem(f,σ,u₀,trange,p2)
-sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true)
-res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint())
-
-
-noise = vec((@. sol.W(tarray)))
-Wextracted = [W[1][1] for W in noise]
-resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp = [resp1, resp2]
-
-@test isapprox(res_p&#39;, resp, rtol = 1e-6)
-# True
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.005&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:tend&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.=-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RKMil&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;interpretation&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:Stratonovich&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res_u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adjoint_sensitivities&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vec&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;isapprox&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# True&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;finish the current backsolve adjoint version,&lt;/li&gt;
 &lt;li&gt;allow for computing the gradients of non-commuting SDEs,&lt;/li&gt;
 &lt;li&gt;implement also an interpolation adjoint version,&lt;/li&gt;
 &lt;li&gt;benchmark it with respect to AD approaches&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
diff --git a/tag/adjoint-sensitivity-methods/page/1/index.html b/tag/adjoint-sensitivity-methods/page/1/index.html
index e3d99c8..b5f16df 100644
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+++ b/tag/adjoint-sensitivity-methods/page/1/index.html
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--- a/tag/analytical-predictors/index.html
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@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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-  <h1>analytical predictors</h1>
+  <h1>Analytical Predictors</h1>
 
   
 
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 <div class="universal-wrapper">
   
 
+
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-    
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-    <div>
-      <h2><a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a></h2>
-      <div class="article-style">
-        
-          Data-driven methods based on sample instances of the state of a physical system as a function of the system's parameters.
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+      <a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a>
+    </div>
 
     
+    <a href="/project/ml_for_pt/"  class="summary-link">
+      <div class="article-style">
+        Data-driven methods based on sample instances of the state of a physical system as a function of the system&rsquo;s parameters.
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+    </a>
     
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+  <span >
+      FS in Collaboration With Julian Arnold</span>, <span >
+      Flemming Holtorf</span>, <span >
+      Eliska Greplova</span>, <span >
+      Agnes Valenti</span>, <span >
+      Martin Zonda</span>, <span >
+      Axel Lode</span>, <span >
+      Gregor Boschung</span>, <span >
+      Sebastian Huber</span>, <span >
+      And Niels Lörch</span>
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+  Code
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.033052" target="_blank" rel="noopener">
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+  
+  
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+  
+  
+  
     
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2311.09128" target="_blank" rel="noopener">
+    NeurIPS Workshop</a>
+
+
+    </div>
     
+
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+  <div class="ml-3">
     
     
+      
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+      <a href="/project/ml_for_pt/" >
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+            alt="Machine Learning for Phase Transitions" loading="lazy">
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+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
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@@ -642,42 +1086,18 @@ <h2><a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a></h
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@@ -686,31 +1106,53 @@ <h2><a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a></h
 
   
   
-  <div class="container">
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+  
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-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
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diff --git a/tag/analytical-predictors/index.xml b/tag/analytical-predictors/index.xml
index c8ccff4..24b41fd 100644
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diff --git a/tag/automatic-differentiation/index.html b/tag/automatic-differentiation/index.html
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-  <title>Automatic Differentiation | FS</title>
 
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+        Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris&rsquo; article3 for a detailed examination.
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         </button>
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diff --git a/tag/automatic-differentiation/index.xml b/tag/automatic-differentiation/index.xml
index 54c1277..b49bd03 100644
--- a/tag/automatic-differentiation/index.xml
+++ b/tag/automatic-differentiation/index.xml
@@ -5,9 +5,9 @@
     <link>https://frankschae.github.io/tag/automatic-differentiation/</link>
       <atom:link href="https://frankschae.github.io/tag/automatic-differentiation/index.xml" rel="self" type="application/rss+xml" />
     <description>Automatic Differentiation</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Sun, 01 Aug 2021 12:03:17 +0200</lastBuildDate>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sun, 01 Aug 2021 12:03:17 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
       <title>Automatic Differentiation</title>
       <link>https://frankschae.github.io/tag/automatic-differentiation/</link>
     </image>
@@ -17,33 +17,17 @@
       <link>https://frankschae.github.io/post/abstract_differentiation/</link>
       <pubDate>Sun, 01 Aug 2021 12:03:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/abstract_differentiation/</guid>
-      <description>&lt;p&gt;
-&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is 
-&lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&#39; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and 
-&lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, 
-&lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different 
-&lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by 
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a 
-&lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
-&lt;p&gt;
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
+      <description>&lt;p&gt;&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is &lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&amp;rsquo; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and &lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, &lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different &lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by &lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a &lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
+&lt;p&gt;&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
 &lt;h2 id=&#34;optimization-of-the-rosenbrock-function&#34;&gt;Optimization of the Rosenbrock function&lt;/h2&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
+&lt;p&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
 &lt;p&gt;$$
 g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2.
 $$&lt;/p&gt;
 &lt;p&gt;The function $g$ has a global minimum at $(x_1^\star, x_2^\star)= (a, a^2)$ with $g(x_1^\star, x_2^\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.&lt;/p&gt;
-&lt;p&gt;Let us now implement the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
+&lt;p&gt;Let us now implement the &lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
 &lt;p&gt;$$
 S({\bf x}) = \frac{1}{2} \sum_{i=1}^M f_i({\bf x})^2.
 $$&lt;/p&gt;
@@ -51,298 +35,275 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 {\bf x}^{k+1} = {\bf x}^k - \alpha_k \left(J^T J \right)^{-1} J^T f({\bf x}^k),
 $$
-where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a 
-&lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
+where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a &lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;The following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\bf x^\star}=(1,1)$ and our initial guess ${\bf x_0}=(0,-0.1)$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Pkg
-path = @__DIR__
-cd(path); Pkg.activate(&amp;quot;.&amp;quot;); Pkg.instantiate()
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@__DIR__&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;activate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;.&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;instantiate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## AbstractDifferentiation is not released yet!!&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AbstractDifferentiation&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Enzyme&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;UnPack&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Rosenbrock function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualization&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt;    
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.6&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.5&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;linealpha&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;colorbar&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;camera&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;heatmap&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_0&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:circle&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:green&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x^\star&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:star&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:red&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Rosenbrock.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-## AbstractDifferentiation is not released yet!!
 
-using AbstractDifferentiation
-using Test, LinearAlgebra
-using FiniteDifferences, ForwardDiff, Zygote
-using Enzyme, UnPack
-using Plots, LaTeXStrings
-# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6
 
-## Rosenbrock function
-# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²
-g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2
 
-# visualization
-p = [1.0,100.0]
-x₀ = [0.0,-0.1]
-xopt = [1.0,1.0]
 
-do_plot = true
-if do_plot    
-    x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5
-    z = Surface((x₁,x₂)-&amp;gt;g([x₁,x₂],p), x₁, x₂)
-    pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true,
-                labelfontsize=20,camera = (3,50),
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
 
-    pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp),
-                labelfontsize=20,
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
-    scatter!(pl2, [(x₀[1],x₀[2])], label=L&amp;quot;x_0&amp;quot;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green)
-    scatter!(pl2, [(xopt[1],xopt[2])],label=L&amp;quot;x^\star&amp;quot;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red)
 
-    pl = plot(pl1,pl2, layout=(2,1))
-    savefig(pl, &amp;quot;Rosenbrock.png&amp;quot;)
-end
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
 
 
-
-
-
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Rosenbrock.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;To apply the Gauss-Newton algorithm to the Rosenbrock function $g$, we first cast $g$ into an appropriate form fulfilling $S({\bf x})$, i.e., we use:&lt;/p&gt;
 &lt;p&gt;$$
 f:\mathbb{R}^2\rightarrow\mathbb{R}^2:  {\bf x} \mapsto \begin{pmatrix}
-f_1({\bf x}) \\&lt;br&gt;
-f_2({\bf x}) \\&lt;br&gt;
+f_1({\bf x}) \\
+f_2({\bf x}) \\
 \end{pmatrix} = \begin{pmatrix}
-\sqrt{2}(a-x_1) \\&lt;br&gt;
-\sqrt{2b}(x_2-x_1^2)\\&lt;br&gt;
+\sqrt{2}(a-x_1) \\
+\sqrt{2b}(x_2-x_1^2)\\
 \end{pmatrix},
 $$&lt;/p&gt;
 &lt;p&gt;instead of $g$. We can easily compute the Jacobian of $f$ manually&lt;/p&gt;
 &lt;p&gt;$$
 J =  \begin{pmatrix}
--\sqrt{2} &amp;amp; 0 \\&lt;br&gt;
--2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\&lt;br&gt;
+-\sqrt{2} &amp;amp; 0 \\
+-2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\
 \end{pmatrix}.
 $$&lt;/p&gt;
 &lt;p&gt;We can then implement a (simple, non-optimized) version of the Gauss-Newton algorithm as follows.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# bring Rosenbrock function into the form &amp;quot;sum of squares of functions&amp;quot;
-f1(x,p) = convert(eltype(x),sqrt(2))*(p[1]-x[1])
-f2(x,p) = convert(eltype(x),sqrt(2*p[2]))*(x[2]-x[1]^2)
-f(x,p) = [f1(x,p),f2(x,p)]
-function f(res,x,p) # Enzyme works with inplace functions
-	res[1] = f1(x,p)
-	res[2] = f2(x,p)
-	return nothing
-end
-
-## manually pre-defined Jacobian
-function Jacobian(x,p)
-  [-convert(eltype(x),sqrt(2))   0
-  -2*x[1]*convert(eltype(x),sqrt(2*p[2]))  convert(eltype(x),sqrt(2*p[2]))]
-end
-
-## Gauss-Newton scheme
-function GaussNewton!(xs, x, p; maxiter=8, backend=nothing)
-    for i=1:maxiter
-        x = step(x, p, backend)
-        @info i
-        @show x
-        push!(xs, x)
-    end
-    return xs, x
-end
-done(x,x2,p) = g(x2,p) &amp;lt; g(x,p)
-function step(x, p, backend::Nothing, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = Jacobian(x,p)
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;xs = [x₀]
-GaussNewton!(xs, x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# output:
-[ Info: 1 ]
-x = [0.125, -0.08750000000000001]
-[ Info: 2 ]
-x = [0.234375, -0.047265625000000006]
-[ Info: 3 ]
-x = [0.4257812499999995, 0.06800537109374968]
-[ Info: 4 ]
-x = [0.5693359374999986, 0.21857223510742047]
-[ Info: 5 ]
-x = [0.784667968749996, 0.5165503501892037]
-[ Info: 6 ]
-x = [0.9999999999999961, 0.9536321163177449]
-[ Info: 7 ]
-x = [0.9999999999999989, 0.9999999999999999]
-[ Info: 8 ]
-x = [1.0, 1.0]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of 
-&lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## FiniteDifferences
-struct FDMBackend{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend() = FDMBackend(central_fdm(5, 1))
-const fdm_backend = FDMBackend()
-# Minimal interface
-AD.@primitive function jacobian(ab::FDMBackend, f, xs...)
-    return FiniteDifferences.jacobian(ab.alg, f, xs...)
-end
-
-# AD Jacobian returns tuple
-# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]
-# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]
-
-@test AD.jacobian(fdm_backend, x-&amp;gt;f(x,p), x₀)[1] ≈ Jacobian(x₀, p)
-@test AD.jacobian(fdm_backend, f, x₀, p)[1] ≈ Jacobian(x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function step(x, p, backend, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = AD.jacobian(backend, f, x, p)[1]
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-
-
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=fdm_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If we want to use reverse-mode AD instead, for example via 
-&lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Zygote
-struct ZygoteBackend &amp;lt;: AD.AbstractReverseMode end
-const zygote_backend = ZygoteBackend()
-AD.@primitive function pullback_function(ab::ZygoteBackend, f, xs...)
-    return function (vs)
-        # Supports only single output
-        _, back = Zygote.pullback(f, xs...)
-        if vs isa AbstractVector
-            back(vs)
-        else
-            @assert length(vs) == 1
-            back(vs[1])
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(zygote_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=zygote_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (
-&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## ForwardDiff
-struct ForwardDiffBackend &amp;lt;: AD.AbstractForwardMode end
-const forwarddiff_backend = ForwardDiffBackend()
-AD.@primitive function pushforward_function(ab::ForwardDiffBackend, f, xs...)
-    # jvp = f&#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0
-    return function (vs)
-        if xs isa Tuple
-            @assert length(xs) &amp;lt;= 2
-            if length(xs) == 1
-                (ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1]),0),)
-            else
-                ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1], xs[2]+h*vs[2]),0)
-            end
-        else
-            ForwardDiff.derivative(h-&amp;gt;f(xs+h*vs),0)
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(forwarddiff_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=forwarddiff_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;where we have used that the Jacobian-vector product $f&#39;(x)v$, i.e., the primitives of forward-mode AD, can be computed by 
-&lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. 
-&lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Enzyme
-struct EnzymeBackend{T1,T2,T3,T4} &amp;lt;: AD.AbstractReverseMode
-    out::T1
-    λ::T2
-    ∂f_∂x::T3
-    ∂f_∂p::T4
-end
-
-out = zero(x₀)
-λ = zero(x₀)
-∂f_∂x = zero(x₀)
-∂f_∂p = zero(p)
-
-const enzyme_backend = EnzymeBackend(out,λ,∂f_∂x,∂f_∂p)
-AD.@primitive function pullback_function(ab::EnzymeBackend, f, xs...)
-    return function (vs)  
-        # enzyme works only with inplace functions
-        if !(vs isa AbstractVector)
-            @assert length(vs) == 1 # Supports only single output
-            vs = vs[1]
-        end
-
-        if xs isa Tuple
-            @assert length(xs) == 2  # hard-coded for use case with two inputs
-            x₀ = xs[1]
-            p = xs[2]
-        end
-
-        @unpack out, λ, ∂f_∂x, ∂f_∂p = ab # cached in the struct, could also be created in here
-
-        ∂f_∂x .*= false
-        ∂f_∂p .*= false
-        out .*= false
-
-        copyto!(λ, vs)
-
-        autodiff(Duplicated(out, λ), Duplicated(x₀, ∂f_∂x), Duplicated(p, ∂f_∂p)) do _out,_x, _p
-            f(_out,_x,_p)
-        end
-        return (∂f_∂x,∂f_∂p)
-    end
-end
-AD.isinplace(ab::EnzymeBackend) = true
-AD.primalvalue(ab::EnzymeBackend, nothing, f, xs) = (f(ab.out,xs...);return ab.out)
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(enzyme_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=enzyme_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# bring Rosenbrock function into the form &amp;#34;sum of squares of functions&amp;#34;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Enzyme works with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## manually pre-defined Jacobian&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;   &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Gauss-Newton scheme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# output:&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.125&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.08750000000000001&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.234375&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.047265625000000006&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.4257812499999995&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.06800537109374968&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5693359374999986&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.21857223510742047&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.784667968749996&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5165503501892037&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;6&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999961&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9536321163177449&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;7&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999989&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9999999999999999&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of &lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## FiniteDifferences&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Minimal interface&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# AD Jacobian returns tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If we want to use reverse-mode AD instead, for example via &lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## ForwardDiff&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractForwardMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c&#34;&gt;# jvp = f&amp;#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;where we have used that the Jacobian-vector product $f&amp;rsquo;(x)v$, i.e., the primitives of forward-mode AD, can be computed by &lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. &lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Enzyme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;  
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# enzyme works only with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;  &lt;span class=&#34;c&#34;&gt;# hard-coded for use case with two inputs&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@unpack&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# cached in the struct, could also be created in here&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;autodiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;do&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;isinplace&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;primalvalue&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
 &lt;h2 id=&#34;some-current-glitches&#34;&gt;Some current glitches&lt;/h2&gt;
 &lt;p&gt;First, the push forward of a tuple of vectors, e.g., $(v_1, v_2)$, for a function with several input arguments is currently ambiguous. While &lt;code&gt;AD.jacobian&lt;/code&gt; primitives and &lt;code&gt;AD.pullback_function&lt;/code&gt; primitives interpret the push forward of our $f$ function as&lt;/p&gt;
 &lt;p&gt;$$
@@ -352,86 +313,77 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\partial f(x_0,p)}{\partial x} v_1 + \frac{\partial f(x_0,p)}{\partial p} v_2.
 $$&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous
-vs = (randn(2), randn(2))
-res1 = AD.pushforward_function(fdm_backend, f, x₀, p)(vs)
-res2 = AD.pushforward_function(forwarddiff_backend, f, x₀, p)(vs)
-
-@test res2 ≈ res1[1] + res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;res2a = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((vs[1], zero(vs[2])))
-res2b = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((zero(vs[1]), vs[2]))
-
-@test res2a ≈ res1[1]
-@test res2b ≈ res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;struct FDMBackend2{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend2() = FDMBackend2(central_fdm(5, 1))
-const fdm_backend2 = FDMBackend2()
-AD.@primitive function pushforward_function(ab::FDMBackend2, f, xs...)
-    return function (vs)
-        FDM.jvp(ab.alg, f, tuple.(xs, vs)...)
-    end
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian only defined with respect to single input variable
-@test_throws AssertionError H1 = AD.hessian(forwarddiff_backend, g, x₀, p)
-H1 = AD.hessian(forwarddiff_backend, x-&amp;gt;g(x,p), x₀)
-H2 = AD.hessian(forwarddiff_backend, p-&amp;gt;g(x₀,p), p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)
-backends = AD.HigherOrderBackend((forwarddiff_backend,zygote_backend))
-H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;FDM&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jvp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tuple&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian only defined with respect to single input variable&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test_throws&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AssertionError&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;HigherOrderBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;There are many other use cases, e.g.,&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;AbstractDifferentiation.jl is by no means complete yet. We are still in the very early stages, but we hope to make significant progress in the coming weeks. Some of the next steps are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;fixing remaining bugs, e.g., with respect to the computation of the Hessian and&lt;/li&gt;
-&lt;li&gt;adding AD/Finite Differentiation packages such as 
-&lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
+&lt;li&gt;adding AD/Finite Differentiation packages such as &lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
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-  <h1>Bose-Hubbard model</h1>
+  <h1>Bose-Hubbard Model</h1>
 
   
 
@@ -552,89 +758,286 @@ <h1>Bose-Hubbard model</h1>
 <div class="universal-wrapper">
   
 
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+
   
   
     
-    
-    
-    <div>
-      <h2><a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a></h2>
-      <div class="article-style">
-        
-          Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
-        
-      </div>
-    </div>
+
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+
   
+
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+
 
   
 
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-      
+<div class="media stream-item view-compact">
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+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a>
+    </div>
 
     
+    <a href="/project/mbdynamics/"  class="summary-link">
+      <div class="article-style">
+        Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
+      </div>
+    </a>
     
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+  <span >
+      FS in Collaboration With Laurent De Forges De Parny</span>, <span >
+      Miguel Bastarrachea</span>, <span >
+      Axel Lode</span>, <span >
+      And Andreas Buchleitner</span>
+      </div>
       
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+      
 
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-        <pre><code class="tex hljs"></code></pre>
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+        <pre><code></code></pre>
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+
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diff --git a/tag/bosonic-systems/index.xml b/tag/bosonic-systems/index.xml
index 98fc03d..5621b87 100644
--- a/tag/bosonic-systems/index.xml
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-      <title>Bosonic systems</title>
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+      <title>Bosonic Systems</title>
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index aeed4f5..517266c 100644
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+<!DOCTYPE html>
+<html lang="en-us">
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+    <title>https://frankschae.github.io/tag/bosonic-systems/</title>
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+</html>
diff --git a/tag/chaotic-systems/index.html b/tag/chaotic-systems/index.html
index bd9fd84..a9615a1 100644
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+<!DOCTYPE html>
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+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
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@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+  
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-  
 
 
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         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
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         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
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+  <div class="page-header header--fixed">
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+        <ul class="navbar-nav d-md-inline-flex">
           
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-          <a class="nav-link " href="/#about"><span>Home</span></a>
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-        
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+            
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+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
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-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
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+          
+          
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+            
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+          
+
+          
+          
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+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
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+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
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+
+          
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+
+          
+        </ul>
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-  <h1>Chaotic systems</h1>
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@@ -552,89 +758,231 @@ <h1>Chaotic systems</h1>
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-      <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></h2>
-      <div class="article-style">
-        
-          In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
-        
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+      <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
+    </div>
 
     
+    <a href="/post/shadowing/"  class="summary-link">
+      <div class="article-style">
+        In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
+      </div>
+    </a>
     
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diff --git a/tag/chaotic-systems/index.xml b/tag/chaotic-systems/index.xml
index 5bb1b68..1d8e7f0 100644
--- a/tag/chaotic-systems/index.xml
+++ b/tag/chaotic-systems/index.xml
@@ -1,14 +1,14 @@
 <?xml version="1.0" encoding="utf-8" standalone="yes" ?>
 <rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
   <channel>
-    <title>Chaotic systems | FS</title>
+    <title>Chaotic Systems | FS</title>
     <link>https://frankschae.github.io/tag/chaotic-systems/</link>
       <atom:link href="https://frankschae.github.io/tag/chaotic-systems/index.xml" rel="self" type="application/rss+xml" />
-    <description>Chaotic systems</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Fri, 02 Jul 2021 11:08:22 +0200</lastBuildDate>
+    <description>Chaotic Systems</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Fri, 02 Jul 2021 11:08:22 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
-      <title>Chaotic systems</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Chaotic Systems</title>
       <link>https://frankschae.github.io/tag/chaotic-systems/</link>
     </image>
     
@@ -18,8 +18,7 @@
       <pubDate>Fri, 02 Jul 2021 11:08:22 +0200</pubDate>
       <guid>https://frankschae.github.io/post/shadowing/</guid>
       <description>&lt;p&gt;In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &amp;ldquo;butterfly effect&amp;rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.&lt;/p&gt;
-&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
+&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:&lt;/p&gt;
 &lt;p&gt;$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -33,57 +32,59 @@ Under the assumption of ergodicity, $\langle g \rangle_∞$ only depends on $p$.
 &lt;p&gt;One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{pmatrix}
-\text{d}x \\&lt;br&gt;
-\text{d}y \\&lt;br&gt;
-\text{d}z \\&lt;br&gt;
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\&lt;br&gt;
-x(\rho-z) - y\\&lt;br&gt;
-x y - \beta z \\&lt;br&gt;
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Statistics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;QuadGK&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SparseArrays&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# simulate 1 trajectory of the Lorenz system forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;28.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;initial&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;attractor&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
 
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
 
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &amp;quot;initial&amp;quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&amp;quot;attractor&amp;quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -93,81 +94,66 @@ savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
 
 
 
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.&lt;/p&gt;
-&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
+&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
 &lt;p&gt;$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function compute_objective(sol)
-  quadgk(t-&amp;gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&amp;quot;z(t)&amp;quot;, xlabel=L&amp;quot;t&amp;quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&amp;quot;\langle z\rangle&amp;quot;, lw = 2)
-savefig(pl2, &amp;quot;zsingle.png&amp;quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
-
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
-
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&amp;quot;\langle z\rangle&amp;quot;, xlabel=L&amp;quot;\rho&amp;quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &amp;quot;zvsrho.png&amp;quot;)
-
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &amp;quot;z.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We obtain:&lt;/p&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;quadgk&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;atol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;mean_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zsingle.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# for each value of the parameter, solve 20 times the initial value problem&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# wrap the procedure inside a function depending on p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ribbon&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;ho&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zvsrho.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;margin&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;600&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;300&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;z.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We obtain:&lt;/p&gt;
 
 
-  
 
 
 
@@ -179,40 +165,35 @@ savefig(pl4, &amp;quot;z.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/z.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/z.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;z.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$&lt;/p&gt;
 &lt;h2 id=&#34;conventional-forward-mode-sensitivity-analysis-and-finite-differencing&#34;&gt;Conventional forward-mode sensitivity analysis and finite-differencing&lt;/h2&gt;
-&lt;p&gt;For non-chaotic systems, we would just use the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we find diverging values:&lt;/p&gt;
+&lt;p&gt;For non-chaotic systems, we would just use the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_forward&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_calculus&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we find diverging values:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\&lt;br&gt;
+&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$&lt;/p&gt;
@@ -220,44 +201,39 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$&lt;/p&gt;
-&lt;p&gt;Similarly, using 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-     xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&amp;quot;Float32&amp;quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
-
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&amp;quot;Float32&amp;quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-
-savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;p&gt;Similarly, using &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;500&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_Floats.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -269,18 +245,14 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Without animation:&lt;/p&gt;
 
@@ -288,72 +260,61 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-  
-
-
-
-
-
 
 
 
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_Floats.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-&lt;p&gt;Luckily, the 
-&lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
+&lt;p&gt;Luckily, the &lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
 &lt;blockquote&gt;
 &lt;p&gt;Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&amp;ldquo;shadows&amp;rdquo;) the numerically computed one.&lt;/p&gt;
 &lt;/blockquote&gt;
 &lt;h2 id=&#34;shadowing-methods&#34;&gt;Shadowing methods&lt;/h2&gt;
-&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref2:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref3:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;lss-and-nilss-for-the-lorenz-system&#34;&gt;LSS and NILSS for the Lorenz system&lt;/h2&gt;
-&lt;p&gt;Switching to LSS or NILSS within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
+&lt;p&gt;Switching to LSS or NILSS within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# objective&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# LSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;CosWindowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0095888187322035&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Cos2Windowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0343951385924328&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0284286902740765&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_adjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.028428690274077&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# select via sensealg in solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9694728321500617&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;&lt;code&gt;CosWindowing()&lt;/code&gt; (default)&lt;/li&gt;
 &lt;li&gt;&lt;code&gt;Cos2Windowing()&lt;/code&gt;&lt;/li&gt;
@@ -362,72 +323,68 @@ dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
 &lt;p&gt;Additionally, an adjoint implementation &lt;code&gt;AdjointLSS()&lt;/code&gt; is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that &lt;code&gt;AdjointLSS(alpha=10.0)&lt;/code&gt; agrees perfectly with &lt;code&gt;ForwardLSS(alpha=10.0)&lt;/code&gt;. In all cases considered, we find the expected sensitivity value of $\approx 1$.&lt;/p&gt;
 &lt;p&gt;However, the use of &lt;code&gt;LSS()&lt;/code&gt; is (typically) much more expensive than the use of &lt;code&gt;NILSS()&lt;/code&gt;, because &lt;code&gt;LSS()&lt;/code&gt; needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of &lt;code&gt;NILSS()&lt;/code&gt; scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making &lt;code&gt;NILSS()&lt;/code&gt; more efficient.&lt;/p&gt;
 &lt;p&gt;In the &lt;code&gt;NILSS()&lt;/code&gt; algorithm, the user can control the number of steps per segment as well as the number of segments.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# NILSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# make sure trajectory is fully on the attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;120.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9966924374966089&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0416028730638789&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Warning: Detected a large value of ξ at the beginning of a segment.&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref4:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;NILSAS&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;li&gt;FD-NILSS&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref4:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019) &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019)&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019) &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019)&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
diff --git a/tag/chaotic-systems/page/1/index.html b/tag/chaotic-systems/page/1/index.html
index 37d8335..258d224 100644
--- a/tag/chaotic-systems/page/1/index.html
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\ No newline at end of file
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diff --git a/tag/convex-optimization/index.html b/tag/convex-optimization/index.html
index 8ba2822..87c6c49 100644
--- a/tag/convex-optimization/index.html
+++ b/tag/convex-optimization/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
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-  <h1>convex optimization</h1>
+  <h1>Convex Optimization</h1>
 
   
 
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 <div class="universal-wrapper">
   
 
+
+  
+
   
   
     
-    
-    
-    <div>
-      <h2><a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a></h2>
-      <div class="article-style">
-        
-          We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
-        
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+      <a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a>
+    </div>
 
     
+    <a href="/project/controlbounds/"  class="summary-link">
+      <div class="article-style">
+        We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
+      </div>
+    </a>
     
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+  <span >
+      FS in Collaboration With Flemming Holtorf</span>, <span >
+      Julian Arnold</span>, <span >
+      Chris Rackauckas</span>, <span >
+      And Alan Edelman</span>
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-        <pre><code class="tex hljs"></code></pre>
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diff --git a/tag/differentiable-programming/index.html b/tag/differentiable-programming/index.html
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-      <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a></h2>
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+        Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
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diff --git a/tag/differentiable-programming/index.xml b/tag/differentiable-programming/index.xml
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+      <title>Differentiable Programming</title>
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index 28e86e6..2163423 100644
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+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
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+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
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+<link rel="canonical" href="https://frankschae.github.io/tag/event-handling/" />
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+  <link rel="manifest" href="/manifest.webmanifest" />
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+<meta name="theme-color" content="#1565c0" />
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+<meta property="og:type" content="website" />
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+    <meta property="og:updated_time" content="2021-07-16T13:24:04&#43;02:00" />
   
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+    <link rel="alternate" href="/tag/event-handling/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2021-07-16T13:24:04&#43;02:00">
+
   
 
+
   
+  <title>Event Handling | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>Event handling | FS</title>
 
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-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
+
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+  
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+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
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+<aside class="search-modal" id="search">
   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
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         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
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         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
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+      
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     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
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+  <div class="page-header header--fixed">
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@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
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+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
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+        <a class="navbar-brand" href="/">FS</a>
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+              
+              
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+                
+              
+              
             
           
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
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+          
 
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+          
+          
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+          
 
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+          
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+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
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-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
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+          
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+                
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+            
+          
 
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+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
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-          <a class="nav-link " href="/#software"><span>Open Source Software</span></a>
-        </li>
+          
 
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+          
 
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+          
+            
+              
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+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
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+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
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+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
@@ -461,65 +681,51 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
+
         
         
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-          
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-            
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-          
         
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-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
+            </a>
+          </div>
         </li>
+        
 
         
         
 
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-    </div>
-
-    <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
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-            <span>Automatic</span>
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-      
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+    </div>
+  </nav>
+</header>
 
-    </ul>
 
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-</nav>
 
+  <div class="page-body">
+    
+    
+    
+
+    
 
-  
 
 
 
@@ -540,7 +746,7 @@ <h1>Search</h1>
     
   
 <div class="universal-wrapper pt-3">
-  <h1>Event handling</h1>
+  <h1>Event Handling</h1>
 
   
 
@@ -552,101 +758,328 @@ <h1>Event handling</h1>
 <div class="universal-wrapper">
   
 
+
+  
+
   
   
     
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+
+
+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a>
+    </div>
+
     
-    
-    <div>
-      <h2><a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a></h2>
+    <a href="/post/bouncing_ball/"  class="summary-link">
       <div class="article-style">
-        
-          In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
-        
+        In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
       </div>
-    </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+        
+
+
+<div class="article-metadata">
+
+  
   
+  
+  
+  <div>
     
+
+  <span >
+      Frank Schäfer and Moritz Schauer</span>
+  </div>
+  
+  
+
+  
+  <span class="article-date">
     
     
-    <div>
-      <h2><a href="/post/hybridde/" >Neural Hybrid Differential Equations</a></h2>
-      <div class="article-style">
-        
-          I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
-        
-      </div>
-    </div>
+      
+    
+    Jul 16, 2021
+  </span>
+  
+
+  
+
+  
+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    19 min read
+  </span>
   
 
   
+  
+  
+  
+
+  
+  
 
 </div>
 
       
+    </div>
 
     
+
+  </div>
+  <div class="ml-3">
     
     
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script>
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+  </div>
+</div>
+
+  
+    
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+  
+
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/hybridde/" >Neural Hybrid Differential Equations</a>
+    </div>
+
+    
+    <a href="/post/hybridde/"  class="summary-link">
+      <div class="article-style">
+        I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
 
-      
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/mermaid/8.4.8/mermaid.min.js" integrity="sha256-lyWCDMnMeZiXRi7Zl54sZGKYmgQs4izcT7+tKc+KUBk=" crossorigin="anonymous" title="mermaid"></script>
       
 
       
         
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+      
     
+    Jun 16, 2021
+  </span>
+  
+
+  
+
+  
+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    7 min read
+  </span>
+  
+
+  
+  
+  
+  
+
+  
+  
+
+</div>
+
+      
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+<script src="/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js"></script>
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@@ -654,42 +1087,18 @@ <h2><a href="/post/hybridde/" >Neural Hybrid Differential Equations</a></h2>
           <p class="search-hit-description">{{snippet}}</p>
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@@ -698,31 +1107,53 @@ <h2><a href="/post/hybridde/" >Neural Hybrid Differential Equations</a></h2>
 
   
   
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@@ -735,7 +1166,8 @@ <h5 class="modal-title">Cite</h5>
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@@ -750,5 +1182,25 @@ <h5 class="modal-title">Cite</h5>
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diff --git a/tag/event-handling/index.xml b/tag/event-handling/index.xml
index 4c33966..cc30432 100644
--- a/tag/event-handling/index.xml
+++ b/tag/event-handling/index.xml
@@ -1,14 +1,14 @@
 <?xml version="1.0" encoding="utf-8" standalone="yes" ?>
 <rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
   <channel>
-    <title>Event handling | FS</title>
+    <title>Event Handling | FS</title>
     <link>https://frankschae.github.io/tag/event-handling/</link>
       <atom:link href="https://frankschae.github.io/tag/event-handling/index.xml" rel="self" type="application/rss+xml" />
-    <description>Event handling</description>
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+    <description>Event Handling</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Fri, 16 Jul 2021 13:24:04 +0200</lastBuildDate>
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-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
-      <title>Event handling</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Event Handling</title>
       <link>https://frankschae.github.io/tag/event-handling/</link>
     </image>
     
@@ -20,96 +20,106 @@
       <description>&lt;p&gt;In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. We highlight differences between explicit&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and implicit events&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;. As a paradigmatic example, we consider a bouncing ball described by the ODE&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\&lt;br&gt;
+\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\
 \text{d}v(t) &amp;amp;= -\mathrm g\thinspace  \text{d}t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;with initial condition&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t=0) &amp;amp;= z_0 = 5, \\&lt;br&gt;
+z(t=0) &amp;amp;= z_0 = 5, \\
 v(t=0) &amp;amp;= v_0 = -0.1.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;The initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.&lt;/p&gt;
-&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
+&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
 &lt;p&gt;When ignoring the bounces, we can straightforwardly integrate the ODE analytically&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\&lt;br&gt;
+z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\
 v(t) &amp;amp;= v_0 - \mathrm g\thinspace  t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;or numerically using the OrdinaryDiffEq package from the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### simulate forward
+&lt;p&gt;or numerically using the OrdinaryDiffEq package from the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### simulate forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# dynamics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# parameters and solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;5.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.9&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward_no_bounce.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
-using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity
-using Plots, LaTeXStrings
 
-# dynamics
-function f(du,u,p,t)
-  du[1] = u[2]
-  du[2] = -p[1]
-end
 
-# parameters and solve
-z0 = 5.0
-v0 = -0.1
-t0 = 0.0
-tend = 1.9
-g = 10
-γ = 0.8
 
-u0 = [z0,v0]
-tspan = (t0,tend)
-p = [g, γ]
-prob = ODEProblem(f,u0,tspan,p)
 
-# plot forward trajectory
-sol = solve(prob,Tsit5(),saveat=0.1)
-pl = plot(sol, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomleft)
-hline!(pl, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34;&gt;&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
+&lt;/p&gt;
 
 
 
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+
+
+
+
+
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward_no_bounce.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Of course, this way the ball continues to fall through the barrier at $z=0$.&lt;/p&gt;
 &lt;h2 id=&#34;forward-simulation-with-events&#34;&gt;Forward simulation with events&lt;/h2&gt;
 &lt;p&gt;At time $\tau$ around $\tau \approx 1$, the ball hits the ground $z(\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\tau$ with new initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau}) &amp;amp;=  z(\tau-) ,\\&lt;br&gt;
+z({\tau}) &amp;amp;=  z(\tau-) ,\\
 v({\tau})&amp;amp;= -\gamma v(\tau-) ,
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -126,15 +136,15 @@ $$&lt;/p&gt;
 &lt;p&gt;The full explicit trajectory $z_{\rm exp}(t) = z(t)$ is determined by&lt;/p&gt;
 &lt;p&gt;$$
 z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\&lt;br&gt;
--0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\
+-0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\
 \quad  - (-0.99005 + t) (-0.99005 \mathrm g + v_0)\gamma  ,&amp;amp; \forall t \ge \tau,
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;where we used&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\&lt;br&gt;
+z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\
 v({\tau})&amp;amp;= -\gamma v({\tau-}) = -\gamma(v_0 - 0.99005 \mathrm g)  .
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -143,15 +153,14 @@ $$&lt;/p&gt;
 a(z,v) = (z, -\gamma v).
 $$&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;DiscreteCallback&lt;/code&gt; in this case to simulate the system.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with DiscreteCallback (explicit event)
-tstar = (v0 + sqrt(v0^2+2*z0*g))/g
-condition1(u,t,integrator) = (t == tstar)
-affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2]
-cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true))
-
-sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar])
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with DiscreteCallback (explicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiscreteCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstops&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
 &lt;h3 id=&#34;implicit-events&#34;&gt;Implicit events&lt;/h3&gt;
 &lt;p&gt;The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)&lt;/p&gt;
 &lt;p&gt;$$
@@ -160,7 +169,7 @@ $$&lt;/p&gt;
 &lt;p&gt;where an event occurs at time $\tau$ if $g(z(\tau),v(\tau),p,\tau) = 0$. We have already used this condition to define our impact time $\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time  $g(z,v,p,t) = 0$.&lt;/p&gt;
 &lt;p&gt;As in the previous case, we can analytically compute the full trajectory of the ball. By substituting the formula for $\tau$ we have at the event time&lt;/p&gt;
 &lt;p&gt;\begin{aligned}
-z({\tau})&amp;amp;= 0, \\&lt;br&gt;
+z({\tau})&amp;amp;= 0, \\
 v({\tau}-)&amp;amp;= - \sqrt{v_0^2 + 2 \mathrm g z_0}
 \end{aligned}&lt;/p&gt;
 &lt;p&gt;for the left limit and&lt;/p&gt;
@@ -170,33 +179,28 @@ v({\tau})&amp;amp;= \gamma \sqrt{v_0^2 + 2 \mathrm g z_0}
 &lt;p&gt;right after the bounce. Thus, the full trajectory $z_{\rm imp}(t) = z(t)$ is given by&lt;/p&gt;
 &lt;p&gt;$$
 (\star) \quad z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\&lt;br&gt;
--\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\
+-\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\
 \quad\cdot \space (-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0} (1 + 2 \gamma)), &amp;amp; \forall t \ge \tau.
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;This is correct even if one substitutes, e.g., a value with higher precision $\mathrm g = 9.81$ for the gravitation constant.&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;ContinuousCallback&lt;/code&gt; in this case.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with ContinuousCallback (implicit event)
-condition2(u,t,integrator) = u[1] # Event happens when condition2(u,t,integrator) == 0
-cb2 = ContinuousCallback(condition2,affect!,save_positions=(true,true))
-sol2 = solve(prob,Tsit5(),callback=cb2,saveat=0.1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# plot forward trajectory
-pl1 = plot(sol1, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;explicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-pl2 = plot(sol2, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;implicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-hline!(pl1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-hline!(pl2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-pl = plot(pl1,pl2)
-savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with ContinuousCallback (implicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Event happens when condition2(u,t,integrator) == 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ContinuousCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;explicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;implicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -208,18 +212,14 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In addition, the implicitly defined impact time via the &lt;code&gt;ContinuousCallback&lt;/code&gt; also changes appropriately when changing the initial conditions or the parameters, for example when using  $\mathrm g = 9.81$ for the gravitation constant. In other words, the event time $\tau=\tau(p,z_0,v_0,t_0)$ is a function of the parameters and initial conditions, and is implicitly defined by the event condition.&lt;/p&gt;
 &lt;p&gt;Suppose we let the ball drop from a somewhat higher position now. Does an increase in height $z$ at $t=0$ give an increase or decrease in height at the end time $t_\text{end}=1.9$? This is something we can answer with sensitivity analysis. For example if we increase the height by (a fraction of) one unit then using $(\star)$&lt;/p&gt;
@@ -228,36 +228,35 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 $$&lt;/p&gt;
 &lt;p&gt;meaning the height at $t_\text{end}$ is also by a corresponding fraction of 0.84 units higher.&lt;/p&gt;
 &lt;p&gt;We can verify this visually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# animate forward trajectory
-sol3 = solve(remake(prob,u0=[u0[1]+0.5,u0[2]]),Tsit5(),callback=cb2,saveat=0.01)
-
-plt2 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;, xlims=(t0,tend))
-hline!(plt2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-plot!(plt2, sol3, tspan=(t0,tend), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)
-
-list_plots = []
-for t in sol3.t
-  tstart = 0.0
-
-  plt1 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;)
-  hline!(plt1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-  plot!(plt1, sol3, tspan=(t0,t), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-  scatter!(plt1,[t,t], sol3(t), color=[1, 2], label=false)
-  plt = plot(plt1,plt2)
-  push!(list_plots, plt)
-end
-
-plot(list_plots[100])
-
-anim = animate(list_plots,every=1)
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# animate forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]]),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -269,18 +268,11 @@ anim = animate(list_plots,every=1)
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/BB.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;The original curve is shown in black in the figure above.&lt;/p&gt;
 &lt;h2 id=&#34;sensitivity-analysis-with-events&#34;&gt;Sensitivity analysis with events&lt;/h2&gt;
@@ -295,7 +287,7 @@ $$&lt;/p&gt;
 &lt;p&gt;Inserting our results for $z(t) = z_{\rm exp}(t)$ instead of $z(t) = z_{\rm imp}(t)$ at $t_{\text{end}}$, a different value for the sensitivity is obtained (a value ignoring the changes in $z$ due to changes in the bouncing time $\tau$), e.g.,&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L^{\text{exp}}}{\text{d} z_0} = \begin{cases}
-1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\&lt;br&gt;
+1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\
 0.84 &amp;amp; \text{with } z(t) = z_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -306,7 +298,7 @@ $$&lt;/p&gt;
 &lt;p&gt;we could be interested in the sensitivity of $L$ with respect $g$. Using the velocity $v(t)$ at the explicit time $t = 0.9905$ instead of the implicit $v(\tau-)$, again a different value for the sensitivity is obtained (a value again ignoring the changes in $\tau$ and not the one we are looking for), even though $\tau = 0.9905$ in both cases:&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L}{\text{d} g} = \begin{cases}
--0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\&lt;br&gt;
+-0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\
 -\frac{g}{\sqrt{v_0^2 + 2 g z_0}} = -0.99995 &amp;amp; \text{ with } v(t) = v_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -323,51 +315,48 @@ $$&lt;/p&gt;
 &lt;p&gt;We write&lt;/p&gt;
 &lt;p&gt;$$\text{solve}(t_0, x_0, t, p)$$&lt;/p&gt;
 &lt;p&gt;$(= x(t))$ for the functional solution of the ODE at time $t$.&lt;/p&gt;
-&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
+&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
 &lt;p&gt;Let us focus on the &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; algorithm which computes the sensitivities&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\&lt;br&gt;
+\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\
 \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}p} &amp;amp;= \lambda_{p}(t_{0}),
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;of a loss function $L$ acting on the final state with respect to the initial state and the parameters. It does so by solving an ODE for $\lambda(s)$ in reverse time from $t$ to $t_0$&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\&lt;br&gt;
+\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\
 \frac{\text{d}\lambda_{p}(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(x(s), \rightarrow p, s)}{\text{d} p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;with initial conditions:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\&lt;br&gt;
+\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\
 \lambda_{p}(t) &amp;amp;= 0.
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The arrows ($\rightarrow$) indicate the variable with respect to which we differentiate, which will become important later when the same variable shows up in multiple function arguments.&lt;/p&gt;
-&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in &lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;&lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, essentially the custom primitive differentiation rule of &lt;code&gt;solve&lt;/code&gt;, is the elementary building block needed to derive sensitivities also in more complicated examples:&lt;/p&gt;
 &lt;p&gt;Consider a loss depending on the state $x(s_i)$ at fix time points $s_i$ through loss functions $L_i^{\text{exp}}$,&lt;/p&gt;
 &lt;p&gt;$$
 L^{\text{exp}} = \sum_i L_i^{\text{exp}}(s_i, x(s_i), p).
 $$&lt;/p&gt;
 &lt;p&gt;Without contortions we can obtain the sensitivity of $L^{\text{exp}}$ in $p$ (or in $x_0$) using the tool we have. For those a bit familiar with automatic differentiation, this is perhaps easiest to see if we write $L^{\text{exp}}$ as pseudo code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1 = solve(t0, x0, s1, p)
-    L = L1(s1, x1, p)
-    x2 = solve(s1, x1, s2, p)
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\&lt;br&gt;
+\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\
 \frac{\text{d}\lambda_{p}(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(x(t), \rightarrow p, t)}{\text{d} p} - \frac{\text{d} L_i^{\text{exp}}(  x(t),\rightarrow p)}{\text{d} p}^\dagger \delta(t-s_i).
 \end{aligned}
 $$&lt;/p&gt;
@@ -381,24 +370,23 @@ $$&lt;/p&gt;
 $$&lt;/p&gt;
 &lt;p&gt;Now we can even properly define the &lt;code&gt;rrule&lt;/code&gt; of &lt;code&gt;solve&lt;/code&gt; in the sense of &lt;code&gt;DiffRules.jl&lt;/code&gt;.&lt;/p&gt;
 &lt;h3 id=&#34;explicit-events-1&#34;&gt;Explicit events&lt;/h3&gt;
-&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1- = solve(t0, x0, s1, p)
-    L = L1-(s1, x1-, p) # saved before affect
-    x1 = a(x1-, p, s1) # affect
-    L += L1(s1, x1, p) # saved after affect
-    x2- = solve(s1, x1, s2, p)
-    L += L2-(s2, x2-, p)
-    x2 = a(x2-, p, s2) # affect
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
+&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref2:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved before affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved after affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\&lt;br&gt;
+\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\
 \lambda_p({s_j}-) &amp;amp;= \lambda_p({s_j}) -  \lambda({s_j})^\dagger \frac{\text{d} a(x({s_j}-), \rightarrow p, {s_j}-)}{\text{d} p}
 \end{aligned}
 $$&lt;/p&gt;
@@ -412,27 +400,26 @@ $$&lt;/p&gt;
 &lt;p&gt;to put emphasis on this, or equivalently, compute for a unspecified function $L(t, x, p)$ the result of $\frac{\text{d}L}{\text{d}p}$ with&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\&lt;br&gt;
+\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\
 &amp;amp;= \frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;which indicates that changing $p$ influences $L$ both through changes in $\tau_1$ as well as changes in&lt;/p&gt;
 &lt;p&gt;$$\xi_1 = x(\tau_1-).$$&lt;/p&gt;
 &lt;p&gt;This case where we have a loss function $L = L_1$ depending on $\tau_1$, $x(\tau_1)$, and $p$
-was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;p&gt;Therefore, the sensitivity of the event time with respect to parameters $\frac{\text{d}\tau}{\text{d}p}$ must be taken into account.
-Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the 
-&lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
+Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the &lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\&lt;br&gt;
+\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The total derivative&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; inside the bracket is:
 $$
 \begin{aligned}
-\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\&lt;br&gt;
+\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;Since&lt;/p&gt;
@@ -455,8 +442,8 @@ for the second term of $\dfrac{\text{d}\tau_1(p)}{\text{d}p}$.&lt;/p&gt;
 &lt;p&gt;We can now write the gradient as:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\&lt;br&gt;
-+&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\&lt;br&gt;
+\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\
++&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\
 +&amp;amp; \frac{\text{d}L(\rightarrow \tau_1(p), \text{solve}(t_0, x_0, \rightarrow \tau_1(p),  p), p)}{\text{d}\tau_1} \frac{\text{d} \tau_1(p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
@@ -525,7 +512,7 @@ $$&lt;/p&gt;
 &lt;li&gt;If both limits contribute to the loss function, the contributions ${\color{red}\lambda_1}$ and ${\color{blue}\lambda_{a,1}}$ are added.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h4 id=&#34;generalization-several-events&#34;&gt;Generalization: several events&lt;/h4&gt;
-&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
+&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
 &lt;p&gt;$$
 \text{solve}(t_0, x_0, t, p)  = \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p),
 $$&lt;/p&gt;
@@ -533,16 +520,15 @@ $$&lt;/p&gt;
 &lt;p&gt;(A good exercise to get familiar with these type of arguments is to verify&lt;/p&gt;
 &lt;p&gt;$$\frac{\text{d}}{\text{d}s} \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p) = 0.)$$&lt;/p&gt;
 &lt;p&gt;Essentially, we know how to address several events already by considering a loss function&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss2(t0, x0, p)
-    tau1, xi1 = solve2(t0, x0, g, p)
-    L = L1(tau1, xi1, p)
-    tau2, xi2 = solve2(tau1, xi1, g, p)
-    L += L2(tau2, xi2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
 &lt;p&gt;Differentiating by hand, we have the following modification of the method from the previous section, which can be derived choosing a particular function $L$ in the previous section which incorporates the &lt;code&gt;solve&lt;/code&gt; from $\tau_1$ to $t_{\text{end}}$. Note&lt;/p&gt;
 &lt;p&gt;$$
 \lambda_{\tau_1}^0 = \frac{\text{d}(\text{solve}(\rightarrow \tau_1,  a(\xi_1,p), t_{\text{end}}, p))}{\text{d} \tau_1} = - f(a(\xi_1,p), p, \tau_1).
@@ -574,43 +560,38 @@ $$&lt;/p&gt;
 &lt;li&gt;Backpropagate $\lambda_J = {\color{red}\lambda_J} + {\color{blue}\lambda_{a,J}} + {\color{green}\lambda_{c,J}}$ to the next event time $\tau_{J-1}$ and iterate over the remaining $J-1$ events.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
-&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
-&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (
-&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$. &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$.&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the &lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
 $$
 \begin{aligned}
-\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\&lt;br&gt;
-&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\&lt;br&gt;
+\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\
+&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\
 &amp;amp;+ \frac{\text{d}f(t, x_1(t),x_2(\rightarrow t),\dots,x_N(t))}{\text{d}t} + \dots +  \frac{\text{d}f(t, x_1(t),x_2(t),\dots,x_N(\rightarrow t))}{\text{d}t}.
 \end{aligned}
-$$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+$$&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -619,21 +600,11 @@ $$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-
       <link>https://frankschae.github.io/post/hybridde/</link>
       <pubDate>Wed, 16 Jun 2021 14:50:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/hybridde/</guid>
-      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
+      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
 &lt;p&gt;This first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).&lt;/p&gt;
-&lt;p&gt;Note: Please check the 
-&lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
+&lt;p&gt;Note: Please check the &lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
 &lt;h2 id=&#34;hybrid-differential-equations&#34;&gt;Hybrid Differential Equations&lt;/h2&gt;
-&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
+&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
 &lt;p&gt;Some relevant examples for hybrid DEs with discrete or continuous callbacks are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; (&lt;code&gt;ContinuousCallback&lt;/code&gt;).&lt;/li&gt;
@@ -643,7 +614,7 @@ $$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-
 &lt;li&gt;pharmacokinetic models&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, where explicit dosing times change the drug concentration in the blood (&lt;code&gt;DiscreteCallback&lt;/code&gt;). The simplest possible example being the one-compartment model.&lt;/li&gt;
 &lt;li&gt;kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (&lt;code&gt;DiscreteCallback&lt;/code&gt;).&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;kicked-harmonic-oscillator&#34;&gt;Kicked harmonic oscillator&lt;/h2&gt;
 &lt;p&gt;Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form&lt;/p&gt;
 &lt;p&gt;$$
@@ -652,40 +623,32 @@ $$&lt;/p&gt;
 &lt;p&gt;where $a=0.1$ and $b=1$ with initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-x(t=0) &amp;amp;= 1  \\&lt;br&gt;
+x(t=0) &amp;amp;= 1  \\
 v(t=0) &amp;amp;= \dot{x}(t=0) = 0.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;This second-order ODE can be 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use 
-&lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity
-using Zygote
-using Random
-u0 = Float32[1.; 0.]
-
-tspan = (0.0f0,50.0f0)
-
-dtsave = 0.5f0
-t = tspan[1]:dtsave:tspan[2]
-
-function oscillator!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  return nothing
-end
-
-prob_data = ODEProblem(oscillator!,u0,tspan)
-
-# ODE without kicks
-pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
+&lt;p&gt;This second-order ODE can be &lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use &lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqFlux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Optim&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# ODE without kicks&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -697,44 +660,36 @@ pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &am
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as &lt;code&gt;1&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;kicktimes = tspan[1]:1:tspan[2]
-function kick!(integrator)
-  integrator.u[end] += one(eltype(integrator.u))
-end
-cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false))
-
-sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t)
-t_data = sol_data.t
-ode_data = Array(sol_data)
-
-# visualize data
-pl1 = plot(t_data,ode_data[1,:],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl1,t_data,ode_data[2,:],label=&amp;quot;data v(t)&amp;quot;)
-
-pl2 = plot(t_data[1:20],ode_data[1,1:20],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl2,t_data[1:20],ode_data[2,1:20],label=&amp;quot;data v(t)&amp;quot;)
-pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;
-
-
-
-
-  
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualize data&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
 
 
@@ -746,63 +701,56 @@ pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting &lt;code&gt;save_positions=(true,true)&lt;/code&gt;, the kicks would be saved before &lt;strong&gt;and&lt;/strong&gt; after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;, we now aim at learning (potentially) missing parts of the model from these data traces.&lt;/p&gt;
 &lt;h3 id=&#34;high-domain-knowledge&#34;&gt;High domain knowledge&lt;/h3&gt;
 &lt;p&gt;For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the &lt;code&gt;affect!&lt;/code&gt; function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of &lt;code&gt;integrator.u&lt;/code&gt;. If we further set the initial value of that component to &lt;code&gt;1&lt;/code&gt;, then the neural network only has to learn that &lt;code&gt;du[3]&lt;/code&gt; is &lt;code&gt;0&lt;/code&gt;. In other words, the output of the neural network must be &lt;code&gt;0&lt;/code&gt; for all states &lt;code&gt;u&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;Random.seed!(123)
-nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1))
-p_nn1 = initial_params(nn1)
-
-function f1!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3] = nn1(u[1:2], p)[1]
-  return nothing
-end
-
-affect!(integrator) = integrator.u[2] += integrator.u[3]
-cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false))
-z0 = Float32[u0;one(u0[1])]
-prob1 = ODEProblem(f1!,z0,tspan,p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# to visualize the predictions of the trained neural network below
-function visualize(prob,p)
-  _prob = remake(prob,p=p)
-  ode_pred = Array(solve(_prob,Tsit5(),callback=cb,
-                 saveat=dtsave))[1:2,:]
-  pl1 = plot(t_data,ode_pred[1,:],label=&amp;quot;x(t)&amp;quot;)
-  scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=&amp;quot;data x(t)&amp;quot;)
-  pl2 = plot(t_data,ode_pred[2,:],label=&amp;quot;v(t)&amp;quot;)
-  scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=&amp;quot;data v(t)&amp;quot;)
-
-  pl = plot(pl1, pl2, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-  return pl, sum(abs2,ode_data .- ode_pred)
-end
-
-pl = plot(solve(prob1,Tsit5(),saveat=t,
-  callback=cb
-  ),label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;123&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FastChain&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tanh&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial_params&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# to visualize the predictions of the trained neural network below&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                 &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -814,76 +762,66 @@ pl = plot(solve(prob1,Tsit5(),saveat=t,
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;untrained_nn.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;which (of course) doesn&amp;rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### loss function
-function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint())
-  _prob = remake(prob,p=p)
-  pred = Array(solve(_prob,Tsit5(),callback=cb,
-               saveat=dtsave,sensealg=sensealg))[1:2,:]
-  sum(abs2,ode_data .- pred)
-end
-
-loss(p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The recently implemented tools are deeply hidden within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., 
-&lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;###################################
-# training loop
-# optimize the parameters for a few epochs with ADAM
-function train(prob, p_nn; sensealg=BacksolveAdjoint())
-  opt = ADAM(0.0003f0)
-  list_plots = []
-  losses = []
-  for epoch in 1:200
-    println(&amp;quot;epoch: $epoch / 200&amp;quot;)
-    _dy, back = Zygote.pullback(p -&amp;gt; loss(p,
-      prob=prob,
-      sensealg=sensealg), p_nn)
-    gs = @time back(one(_dy))[1]
-    push!(losses, _dy)
-    if epoch % 10 == 0
-      # plot every xth epoch
-      pl, test_loss = visualize(prob, p_nn)
-      println(&amp;quot;Loss (epoch: $epoch): $test_loss&amp;quot;)
-      display(pl)
-      push!(list_plots, pl)
-    end
-    Flux.Optimise.update!(opt, p_nn, gs)
-    println(&amp;quot;&amp;quot;)
-  end
-  return losses, list_plots
-end
-
-# plot training loss
-losses, list_plots = train(prob1, p_nn1)
-pl1 = plot(losses, lw = 1.5, xlabel = &amp;quot;epoch&amp;quot;, ylabel=&amp;quot;loss&amp;quot;, legend=false)
-pl2 = list_plots[end]
-pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t,
-   callback=cb
-  ), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-
-pl = plot(pl2,pl3)
-&lt;/code&gt;&lt;/pre&gt;
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### loss function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ReverseDiffAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;               &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The recently implemented tools are deeply hidden within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., &lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;###################################&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# training loop&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# optimize the parameters for a few epochs with ADAM&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ADAM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0003f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;200&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt; / 200&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;%&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;c&#34;&gt;# plot every xth epoch&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_loss&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Loss (epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;): &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$test_loss&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;display&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Optimise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;update!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot training loss&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;epoch&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;loss&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
-  
 
 
 
@@ -895,98 +833,87 @@ pl = plot(pl2,pl3)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/trained1.png&#34; &gt;
 
-
-  &lt;img src=&#34;https://frankschae.github.io/img/trained1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;trained1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We see the expected constant value of &lt;code&gt;u[3]&lt;/code&gt;, indicating a kick to the velocity of &lt;code&gt;+=1&lt;/code&gt;, at the kicking times over the full time interval.&lt;/p&gt;
 &lt;h2 id=&#34;reducing-the-domain-knowledge&#34;&gt;Reducing the domain knowledge&lt;/h2&gt;
-&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
+&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;changing the initial condition of the third component of &lt;code&gt;u&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;using another affect function &lt;code&gt;affect!(integrator) = integrator.u[2] = integrator.u[3]&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;dropping the knowledge that only &lt;code&gt;u[2]&lt;/code&gt; gets a kick by using a neural network with &lt;code&gt;2&lt;/code&gt; outputs (+ a fourth component in the ODE):&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4]
-function f2!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;fitting the parameters $a$ and $b$ simultaneously:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f3!(du,u,p,t)
-  a = p[end-1]
-  b = p[end]
-  nn_weights = p[1:end-2]
-
-  du[1] = u[2]
-  du[2] = -b*u[1] - a*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f3!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;inferring the entire underlying dynamics using a neural network with &lt;code&gt;4&lt;/code&gt; outputs:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f4!(du,u,p,t)
-  Ω = nn3(u[1:2], p)
-
-  du[1] = Ω[1]
-  du[2] = Ω[2]
-  du[3:4] .= Ω[3:4]
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f4!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;etc.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;refine the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;) and&lt;/li&gt;
-&lt;li&gt;support direct usage through the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
+&lt;li&gt;support direct usage through the &lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021) &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021)&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
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+<div class="universal-wrapper pt-3">
+  <h1>Exact Diagonalization</h1>
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+<div class="media stream-item view-compact">
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+      <a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a>
+    </div>
+
+    
+    <a href="/project/mbdynamics/"  class="summary-link">
+      <div class="article-style">
+        Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
+      </div>
+    </a>
+    
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+
+      
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+  <span >
+      FS in Collaboration With Laurent De Forges De Parny</span>, <span >
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+      And Andreas Buchleitner</span>
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+            alt="Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well " loading="lazy">
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+      <a href="/publication/entropy/" >F. Schäfer, M. A. Bastarrachea-Magnani, A. U. J. Lode, L. de Forges de Parny, and A. Buchleitner, Entropy 22, 382 (2020)</a>
+    </div>
+
+    
+    <a href="/publication/entropy/"  class="summary-link">
+      <div class="article-style">
+        We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional …
+      </div>
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+
+      
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+  PDF
+</a>
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+
 
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-      <h2><a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a></h2>
-      <div class="article-style">
-        
-          Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
-        
-      </div>
-    </div>
+  
   
     
-    
-    
-    <div>
-      <h2><a href="/publication/entropy/" >F. Schäfer, M. A. Bastarrachea-Magnani, A. U. J. Lode, L. de Forges de Parny, and A. Buchleitner, Entropy 22, 382 (2020)</a></h2>
-      <div class="article-style">
-        
-          We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact …
-        
-      </div>
-    </div>
   
 
   
 
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+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
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+  <h1>Forward Sensitivity Methods</h1>
 
   
 
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+
   
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-      <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></h2>
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-        
-          In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
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+      <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
+    </div>
 
     
+    <a href="/post/shadowing/"  class="summary-link">
+      <div class="article-style">
+        In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
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@@ -642,42 +990,18 @@ <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensit
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@@ -686,32 +1010,54 @@ <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensit
 
   
   
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diff --git a/tag/forward-sensitivity-methods/index.xml b/tag/forward-sensitivity-methods/index.xml
index f0ceca9..05f2f66 100644
--- a/tag/forward-sensitivity-methods/index.xml
+++ b/tag/forward-sensitivity-methods/index.xml
@@ -1,14 +1,14 @@
 <?xml version="1.0" encoding="utf-8" standalone="yes" ?>
 <rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
   <channel>
-    <title>Forward sensitivity methods | FS</title>
+    <title>Forward Sensitivity Methods | FS</title>
     <link>https://frankschae.github.io/tag/forward-sensitivity-methods/</link>
       <atom:link href="https://frankschae.github.io/tag/forward-sensitivity-methods/index.xml" rel="self" type="application/rss+xml" />
-    <description>Forward sensitivity methods</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Fri, 02 Jul 2021 11:08:22 +0200</lastBuildDate>
+    <description>Forward Sensitivity Methods</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Fri, 02 Jul 2021 11:08:22 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
-      <title>Forward sensitivity methods</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Forward Sensitivity Methods</title>
       <link>https://frankschae.github.io/tag/forward-sensitivity-methods/</link>
     </image>
     
@@ -18,8 +18,7 @@
       <pubDate>Fri, 02 Jul 2021 11:08:22 +0200</pubDate>
       <guid>https://frankschae.github.io/post/shadowing/</guid>
       <description>&lt;p&gt;In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &amp;ldquo;butterfly effect&amp;rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.&lt;/p&gt;
-&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
+&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:&lt;/p&gt;
 &lt;p&gt;$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -33,57 +32,59 @@ Under the assumption of ergodicity, $\langle g \rangle_∞$ only depends on $p$.
 &lt;p&gt;One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{pmatrix}
-\text{d}x \\&lt;br&gt;
-\text{d}y \\&lt;br&gt;
-\text{d}z \\&lt;br&gt;
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\&lt;br&gt;
-x(\rho-z) - y\\&lt;br&gt;
-x y - \beta z \\&lt;br&gt;
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Statistics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;QuadGK&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SparseArrays&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# simulate 1 trajectory of the Lorenz system forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;28.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;initial&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;attractor&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
 
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
 
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &amp;quot;initial&amp;quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&amp;quot;attractor&amp;quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -93,81 +94,66 @@ savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
 
 
 
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.&lt;/p&gt;
-&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
+&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
 &lt;p&gt;$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function compute_objective(sol)
-  quadgk(t-&amp;gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&amp;quot;z(t)&amp;quot;, xlabel=L&amp;quot;t&amp;quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&amp;quot;\langle z\rangle&amp;quot;, lw = 2)
-savefig(pl2, &amp;quot;zsingle.png&amp;quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
-
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
-
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&amp;quot;\langle z\rangle&amp;quot;, xlabel=L&amp;quot;\rho&amp;quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &amp;quot;zvsrho.png&amp;quot;)
-
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &amp;quot;z.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We obtain:&lt;/p&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;quadgk&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;atol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;mean_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zsingle.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# for each value of the parameter, solve 20 times the initial value problem&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# wrap the procedure inside a function depending on p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ribbon&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;ho&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zvsrho.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;margin&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;600&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;300&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;z.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We obtain:&lt;/p&gt;
 
 
-  
 
 
 
@@ -179,40 +165,35 @@ savefig(pl4, &amp;quot;z.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/z.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/z.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;z.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$&lt;/p&gt;
 &lt;h2 id=&#34;conventional-forward-mode-sensitivity-analysis-and-finite-differencing&#34;&gt;Conventional forward-mode sensitivity analysis and finite-differencing&lt;/h2&gt;
-&lt;p&gt;For non-chaotic systems, we would just use the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we find diverging values:&lt;/p&gt;
+&lt;p&gt;For non-chaotic systems, we would just use the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_forward&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_calculus&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we find diverging values:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\&lt;br&gt;
+&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$&lt;/p&gt;
@@ -220,44 +201,39 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$&lt;/p&gt;
-&lt;p&gt;Similarly, using 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-     xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&amp;quot;Float32&amp;quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
-
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&amp;quot;Float32&amp;quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-
-savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;p&gt;Similarly, using &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;500&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_Floats.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -269,18 +245,14 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Without animation:&lt;/p&gt;
 
@@ -288,72 +260,61 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-  
-
-
-
-
-
 
 
 
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_Floats.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-&lt;p&gt;Luckily, the 
-&lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
+&lt;p&gt;Luckily, the &lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
 &lt;blockquote&gt;
 &lt;p&gt;Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&amp;ldquo;shadows&amp;rdquo;) the numerically computed one.&lt;/p&gt;
 &lt;/blockquote&gt;
 &lt;h2 id=&#34;shadowing-methods&#34;&gt;Shadowing methods&lt;/h2&gt;
-&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref2:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref3:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;lss-and-nilss-for-the-lorenz-system&#34;&gt;LSS and NILSS for the Lorenz system&lt;/h2&gt;
-&lt;p&gt;Switching to LSS or NILSS within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
+&lt;p&gt;Switching to LSS or NILSS within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# objective&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# LSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;CosWindowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0095888187322035&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Cos2Windowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0343951385924328&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0284286902740765&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_adjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.028428690274077&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# select via sensealg in solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9694728321500617&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;&lt;code&gt;CosWindowing()&lt;/code&gt; (default)&lt;/li&gt;
 &lt;li&gt;&lt;code&gt;Cos2Windowing()&lt;/code&gt;&lt;/li&gt;
@@ -362,72 +323,68 @@ dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
 &lt;p&gt;Additionally, an adjoint implementation &lt;code&gt;AdjointLSS()&lt;/code&gt; is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that &lt;code&gt;AdjointLSS(alpha=10.0)&lt;/code&gt; agrees perfectly with &lt;code&gt;ForwardLSS(alpha=10.0)&lt;/code&gt;. In all cases considered, we find the expected sensitivity value of $\approx 1$.&lt;/p&gt;
 &lt;p&gt;However, the use of &lt;code&gt;LSS()&lt;/code&gt; is (typically) much more expensive than the use of &lt;code&gt;NILSS()&lt;/code&gt;, because &lt;code&gt;LSS()&lt;/code&gt; needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of &lt;code&gt;NILSS()&lt;/code&gt; scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making &lt;code&gt;NILSS()&lt;/code&gt; more efficient.&lt;/p&gt;
 &lt;p&gt;In the &lt;code&gt;NILSS()&lt;/code&gt; algorithm, the user can control the number of steps per segment as well as the number of segments.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# NILSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# make sure trajectory is fully on the attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;120.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9966924374966089&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0416028730638789&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Warning: Detected a large value of ξ at the beginning of a segment.&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref4:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;NILSAS&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;li&gt;FD-NILSS&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref4:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019) &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019)&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019) &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019)&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
diff --git a/tag/forward-sensitivity-methods/page/1/index.html b/tag/forward-sensitivity-methods/page/1/index.html
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+<!DOCTYPE html>
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@@ -552,190 +758,482 @@ <h1>GSoC 2020</h1>
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+
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+      <a href="/post/gsoc-2020/" >High weak order solvers and adjoint sensitivity analysis for stochastic differential equations</a>
+    </div>
+
     
-    
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-      <h2><a href="/post/gsoc-2020/" >High weak order solvers and adjoint sensitivity analysis for stochastic differential equations</a></h2>
+    <a href="/post/gsoc-2020/"  class="summary-link">
       <div class="article-style">
-        
-          Project summary In this project, we have implemented new promising tools within the SciML organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
-        
+        Project summary In this project, we have implemented new promising tools within the SciML organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
       </div>
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-  
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-    
-    
-    <div>
-      <h2><a href="/post/high-weak/" >High weak order SDE solvers</a></h2>
-      <div class="article-style">
-        
-          This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
-After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
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-      <h2><a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a></h2>
-      <div class="article-style">
-        
-          First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
-        
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-    </div>
+    Aug 26, 2020
+  </span>
   
 
   
 
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+        This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
+After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
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+    <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/"  class="summary-link">
+      <div class="article-style">
+        First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
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         </button>
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diff --git a/tag/gsoc-2020/index.xml b/tag/gsoc-2020/index.xml
index a244f19..73f1e91 100644
--- a/tag/gsoc-2020/index.xml
+++ b/tag/gsoc-2020/index.xml
@@ -5,9 +5,9 @@
     <link>https://frankschae.github.io/tag/gsoc-2020/</link>
       <atom:link href="https://frankschae.github.io/tag/gsoc-2020/index.xml" rel="self" type="application/rss+xml" />
     <description>GSoC 2020</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Wed, 26 Aug 2020 22:59:49 +0200</lastBuildDate>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Wed, 26 Aug 2020 22:59:49 +0200</lastBuildDate>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
       <title>GSoC 2020</title>
       <link>https://frankschae.github.io/tag/gsoc-2020/</link>
     </image>
@@ -18,152 +18,100 @@
       <pubDate>Wed, 26 Aug 2020 22:59:49 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc-2020/</guid>
       <description>&lt;h2 id=&#34;project-summary&#34;&gt;Project summary&lt;/h2&gt;
-&lt;p&gt;In this project, we have implemented new promising tools within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
+&lt;p&gt;In this project, we have implemented new promising tools within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which are relevant for tasks such as optimal control or parameter estimation for stochastic differential equations.
 The high weak order solvers will allow for massive performance advantages for fitting expectations of equations.
 Instead of automatic differentiation (AD) through the operations of an SDE solver, which scales poorly in memory, one can now use efficient stochastic adjoint sensitivity methods.&lt;/p&gt;
 &lt;h2 id=&#34;blog-posts&#34;&gt;Blog posts&lt;/h2&gt;
 &lt;p&gt;The following posts describe the work during the entire period in more detail:&lt;/p&gt;
 &lt;ol&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GSoC 2020: High weak order SDE solvers and their utility in neural SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/high-weak/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;High weak order SDE solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GSoC 2020: High weak order SDE solvers and their utility in neural SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/high-weak/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;High weak order SDE solvers&lt;/a&gt;&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;docs&#34;&gt;Docs&lt;/h2&gt;
-&lt;p&gt;The documentation of the solvers is available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/solvers/sde_solve/#High-Weak-Order-Methods&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.
-Docs with respect to the adjoint sensitivity tools will be available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The documentation of the solvers is available &lt;a href=&#34;https://diffeq.sciml.ai/latest/solvers/sde_solve/#High-Weak-Order-Methods&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.
+Docs with respect to the adjoint sensitivity tools will be available &lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;here&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;achievements&#34;&gt;Achievements&lt;/h2&gt;
 &lt;p&gt;Please find below a list of the PRs carried out during GSoC in the different repositories in chronological order within the SciML ecosystem.&lt;/p&gt;
 &lt;h4 id=&#34;stochasticdiffeqjl&#34;&gt;StochasticDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/285&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Inplace version of DRI1 scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/289&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI1 method&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;tstop fixes for reverse time propagation&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes for EulerHeun scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/327&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Speed up the tests for the DRI1 and RI1 schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/328&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/329&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RDI1WM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/332&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adaptive version of the stochastic Runge-Kutta schemes by embedding&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/333&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RS1 and RS2 schemes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/334&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PL1WM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/337&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NON scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/338&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for weak methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/342&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stochastic improved and modified Euler methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/343&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;COM scheme&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/348&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Computationally more efficient NON variant (NON2)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/285&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Inplace version of DRI1 scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/289&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI1 method&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;tstop fixes for reverse time propagation&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes for EulerHeun scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/327&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Speed up the tests for the DRI1 and RI1 schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/328&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RI3, RI5, RI6, RDI2WM, RDI3WM, and RDI4WM schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/329&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RDI1WM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/332&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adaptive version of the stochastic Runge-Kutta schemes by embedding&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/333&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;RS1 and RS2 schemes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/334&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PL1WM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/337&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NON scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/338&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for weak methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/342&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stochastic improved and modified Euler methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/343&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;COM scheme&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/348&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Computationally more efficient NON variant (NON2)&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/288&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Static array tests&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Levy area for non-commutative noise processes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/288&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Static array tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Levy area for non-commutative noise processes&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqsensitivityjl&#34;&gt;DiffEqSensitivity.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/235&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint sensitivities for steady states&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/237&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Concrete_solve dispatch for steady state problem&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/242&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;BacksolveAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/256&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GPU savety for SDE BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/258&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Tests for concrete solve with respect to SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/260&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Alternative differentiation choices (vjps) for noise Jacobian&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/265&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes and tests for inplace formulation of BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/268&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Efficient SDE BacksolveAdjoint for scalar noise&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/275&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/295&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;InterpolatingAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/298&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for backsolve, steadystate and interpolation adjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/299&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for more general noise processes: replace NoiseGrid by NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/303&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Cheaper non-diagonal noise tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/235&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint sensitivities for steady states&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/237&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Concrete_solve dispatch for steady state problem&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/242&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;BacksolveAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/256&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;GPU savety for SDE BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/258&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Tests for concrete solve with respect to SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/260&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Alternative differentiation choices (vjps) for noise Jacobian&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/265&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes and tests for inplace formulation of BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/268&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Efficient SDE BacksolveAdjoint for scalar noise&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/275&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the SDE Adjoint for non-diagonal noise processes and diagonal noise processes with mixing terms&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/295&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;InterpolatingAdjoint for SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/298&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Citations for backsolve, steadystate and interpolation adjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/299&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for more general noise processes: replace NoiseGrid by NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/303&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Checkpointing fix for BacksolveAdjoint in case of ODEs and SDEs&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/305&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Cheaper non-diagonal noise tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support adjoints for SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support adjoints for SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqnoiseprocessjl&#34;&gt;DiffEqNoiseProcess.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/48&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Multi-dimensional Brownian motion tests&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/49&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Bug fix for inplace form of NoiseGrid&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/51&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversible NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/53&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Relax the size constraints of the available noise processes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/54&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the real-valued white noise process function&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/55&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix of an extraction issue with NoiseGrid&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Extraction and endpoint fixes on NoiseWrapper&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversal of SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/48&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Multi-dimensional Brownian motion tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/49&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Bug fix for inplace form of NoiseGrid&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/51&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversible NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/53&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Relax the size constraints of the available noise processes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/54&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Generalization of the real-valued white noise process function&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/55&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix of an extraction issue with NoiseGrid&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow NoiseWrapper to start at user-specified time points for interpolating parts of a trajectory&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/56&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Extraction and endpoint fixes on NoiseWrapper&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Reversal of SDEs written in the Ito sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqgpujl&#34;&gt;DiffEqGPU.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl/pull/59&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Memory efficient reduction function for ensemble problems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl/pull/59&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Memory efficient reduction function for ensemble problems&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;modelingtoolkitjl&#34;&gt;ModelingToolkit.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/317&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;modelingtoolkitize for SDESystem with conversion function between Ito and Stratonovich sense&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqdevtoolsjl&#34;&gt;DiffEqDevTools.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NoiseWrapper alternative for analyticless convergence tests of SDE solvers&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/74&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;test_convergence() dispatch for ensemble simulations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Work precision set for ensemble problems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/62&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NoiseWrapper alternative for analyticless convergence tests of SDE solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/74&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;test_convergence() dispatch for ensemble simulations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Work precision set for ensemble problems&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqbasejl&#34;&gt;DiffEqBase.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqBase.jl/pull/503&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix concrete_solve tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqBase.jl/pull/503&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix concrete_solve tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;future-work&#34;&gt;Future work&lt;/h2&gt;
 &lt;p&gt;There is still a lot that we&amp;rsquo;d like to do, e.g.,&lt;/p&gt;
@@ -177,12 +125,8 @@ Docs with respect to the adjoint sensitivity tools will be available
 &lt;/ul&gt;
 &lt;p&gt;Contributions, suggestions &amp;amp; comments are always welcome! You might like to join our slac channels #diffeq-bridged and #neuralsde to get in touch.&lt;/p&gt;
 &lt;h2 id=&#34;acknowledgement&#34;&gt;Acknowledgement&lt;/h2&gt;
-&lt;p&gt;I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback.
-I would also like to thank 
-&lt;a href=&#34;https://www.quantumtheory-bruder.physik.unibas.ch/people.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts.
+&lt;p&gt;I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their amazing support during this project. It was a great opportunity to work in such an inspiring collaboration and I highly appreciate their detailed feedback.
+I would also like to thank &lt;a href=&#34;https://www.quantumtheory-bruder.physik.unibas.ch/people.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, Niels Lörch, Martin Koppenhöfer, and Michal Kloc for helpful comments on my blog posts.
 Many thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
 </description>
     </item>
@@ -192,11 +136,8 @@ Many thanks to the very supportive julia community and to Google&amp;rsquo;s ope
       <link>https://frankschae.github.io/post/high-weak/</link>
       <pubDate>Mon, 17 Aug 2020 14:46:35 +0200</pubDate>
       <guid>https://frankschae.github.io/post/high-weak/</guid>
-      <description>&lt;p&gt;This post summarizes our new 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
-for the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
+      <description>&lt;p&gt;This post summarizes our new &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
+for the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
 &lt;p&gt;After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
 Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics.
 Finally, we discuss how to run simulations on GPU hardware.&lt;/p&gt;
@@ -227,16 +168,14 @@ $$
 \rm{E}(|X(T) -Y(T)|) \leq C \cdot h^p,
 $$
 for each $h \in [0, \delta]$.&lt;/p&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
+&lt;p&gt;The &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
 for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms.
 For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI)
 can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where &lt;em&gt;b&lt;/em&gt; has only entries on its diagonal or $m=1$.
 The main difficulty for the construction of strong methods with an order &amp;gt; 1/2 arises from the need of an accurate estimation of
 multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of &lt;em&gt;dW&lt;/em&gt; in the case
 of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed
-in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also 
-&lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
+in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also &lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;weak-convergence&#34;&gt;Weak convergence&lt;/h2&gt;
 &lt;p&gt;Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt; in many situations. In these cases, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient and &amp;ndash; due to the less restrictive formulation of the objective &amp;ndash; those solvers are computationally cheaper than their strong counterparts.
 For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics.
@@ -251,19 +190,17 @@ $~$&lt;/p&gt;
 &lt;p&gt;for any polynomial $g$ for each $h \in [0, \delta]$.&lt;/p&gt;
 &lt;p&gt;We demonstrate below that &lt;strong&gt;high weak order solvers&lt;/strong&gt; are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.&lt;/p&gt;
 &lt;h2 id=&#34;new-high-weak-order-methods&#34;&gt;New high weak order methods&lt;/h2&gt;
-&lt;p&gt;A list of all new weak solvers is available in the 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
+&lt;p&gt;A list of all new weak solvers is available in the &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
 Note that we also implemented methods designed for the Stratonovich sense.
 For the subsequent examples regarding Ito SDEs, we use only a subset of the plethora of second-order weak solvers.
-We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
-We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
+We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
+We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref2:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref3:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
 The latter is the simplest weak solver, where the Gaussian increments of the strong Euler-Maruyama scheme are replaced by
 two-point distributed random variables with similar moment properties.&lt;/p&gt;
 &lt;p&gt;Rößler&amp;rsquo;s SRK schemes are particularly designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;, since only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn and since the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.
 &lt;code&gt;PL1WM()&lt;/code&gt; in contrast needs to simulate &lt;code&gt;m(m+1)/2&lt;/code&gt; random variables but a smaller number of order conditions needs to be fulfilled.&lt;/p&gt;
 &lt;h3 id=&#34;convergence-tests&#34;&gt;Convergence tests&lt;/h3&gt;
-&lt;p&gt;As in the 
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As in the &lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -274,63 +211,61 @@ $$ ~$$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;for the function $f(x)=(x_1)^2$, which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e6&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-using Test
-using Random
-using Plots
-using DiffEqDevTools
-
-function prob_func(prob, i, repeat)
-    remake(prob,seed=seeds[i])
-end
-
-u₀ = [1.0,1.0]
-function f1!(du,u,p,t)
-  @inbounds begin
-    du[1] = -273//512*u[1]
-    du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-  end
-  return nothing
-end
-function g1!(du,u,p,t)
-  @inbounds begin
-    du[1,1] = 1//4*u[1]
-    du[1,2] = 1//16*u[1]
-    du[2,1] = (1-2*sqrt(2))/4*u[1]
-    du[2,2] = 1//10*u[1]+1//16*u[2]
-  end
-  return nothing
-end
-dts = 1 .//2 .^(3: -1:0)
-tspan = (0.0,3.0)
-
-h2(z) = z^2 # but apply it only to u[1]
-
-prob = SDEProblem(f1!,g1!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-numtraj = Int(1e6)
-seed = 100
-Random.seed!(seed)
-seeds = rand(UInt, numtraj)
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h2(sol[end][1]),false),
-        prob_func = prob_func
-        )
-
-sim = test_convergence(dts,ensemble_prob,DRI1(),
-  save_everystep=false,trajectories=numtraj,save_start=false,adaptive=false,
-  weak_timeseries_errors=false,weak_dense_errors=false,
-  expected_value=exp(-3.0)
-  )
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqDevTools&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# but apply it only to u[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Int&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_convergence&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
 Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function for the other aforementioned solvers, we obtain the convergence plot:&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -342,18 +277,12 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/weak_conv.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/weak_conv.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;weak_conv.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Note that the &lt;code&gt;SimplifiedEM&lt;/code&gt; and the &lt;code&gt;EM&lt;/code&gt; scheme fall on top of each other.
 &lt;code&gt;DRI1()&lt;/code&gt; achieves the smallest errors for a fixed &lt;code&gt;dt&lt;/code&gt; in this study.&lt;/p&gt;
@@ -361,35 +290,29 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 &lt;p&gt;Ultimately, we are not only interested in the general convergence slope of an algorithm but also in its speed.
 We&amp;rsquo;d like to select the fastest method depending on the permitted tolerance.
 This is commonly studied using work-precision diagram.
-Thanks to 
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;reltols = 1.0 ./ 4.0 .^ (1:4)
-abstols = reltols#[0.0 for i in eachindex(reltols)]
-setups = [
-          Dict(:alg=&amp;gt;DRI1(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;PL1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;EM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;SimplifiedEM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI2WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false)
-          ]
-test_dt = 1//10000
-appxsol_setup = Dict(:alg=&amp;gt;EM(), :dt=&amp;gt;test_dt)
-wp = @time WorkPrecisionSet(ensemble_prob,
-                            abstols,reltols,setups,test_dt;
-                            maxiters = 1e7,verbose=false,
-                            save_everystep=false,save_start=false,
-                            appxsol_setup = appxsol_setup,
-                            expected_value=exp(-3.0),
-                            trajectories=numtraj, error_estimate=:weak_final)
-plt = plot(wp;legend=:bottomleft)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
+Thanks to &lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;./&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;4.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;c&#34;&gt;#[0.0 for i in eachindex(reltols)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;PL1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;SimplifiedEM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI2WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10000&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WorkPrecisionSet&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;maxiters&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;verbose&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;error_estimate&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:weak_final&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-  
 
 
 
@@ -401,88 +324,87 @@ plt = plot(wp;legend=:bottomleft)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;WorkPrecision.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Therefore, &lt;code&gt;DRI1&lt;/code&gt; has the best performance in this non-commutative noise case if the error is supposed to stay below 1e-3.
 For larger permitted errors, the &lt;code&gt;SimplifiedEM&lt;/code&gt; scheme might be a good choice. However, the first order methods
 are outclassed when high precision is more of a concern.
-We plan to perform more in-depth benchmarks in the near future what will be reported on the 
-&lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
+We plan to perform more in-depth benchmarks in the near future what will be reported on the &lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
 &lt;h3 id=&#34;adaptive-step-size-control&#34;&gt;Adaptive step-size control&lt;/h3&gt;
-&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
+&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
 &lt;p&gt;To use adaptive step-size control, it is sufficient to set &lt;code&gt;adaptive=true&lt;/code&gt; (default setting). Optionally, one may also pass absolute and relative tolerances.&lt;/p&gt;
 &lt;p&gt;The following julia code simulates the stochastic version of the Brusselator equations with intitial condition&lt;/p&gt;
 &lt;p&gt;$$ X(t=0)=  \begin{pmatrix} 0.1 \\ 0\end{pmatrix},$$&lt;/p&gt;
 &lt;p&gt;on a time span $\mathbb{I}=[0, 100]$ for adaptive (&lt;code&gt;sol&lt;/code&gt;) and fixed step sizes (&lt;code&gt;sol_na&lt;/code&gt;):&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq, DiffEqNoiseProcess, Random
-using Plots
-using DiffEqGPU
-
-function prob_func(prob, i, repeat)
-  Random.seed!(seeds[i])
-  W = WienerProcess(0.0,0.0,0.0)
-  remake(prob,noise=W)
-end
-
-function brusselator_f!(du,u,p,t)
-  @inbounds begin
-    du[1] = (p[1]-1)*u[1]+p[1]*u[1]^2+(u[1]+1)^2*u[2]
-    du[2] = -p[1]*u[1]-p[1]*u[1]^2-(u[1]+1)^2*u[2]
-  end
-  nothing
-end
-
-function scalar_noise!(du,u,p,t)
-  @inbounds begin
-    du[1] = p[2]*u[1]*(1+u[1])
-    du[2] = -p[2]*u[1]*(1+u[1])
-   end
-   nothing
-end
-
-
-# fix seeds
-seed = 100
-Random.seed!(seed)
-numtraj= 100
-seeds = rand(UInt, numtraj)
-W = WienerProcess(0.0,0.0,0.0)
-
-
-u0 = [-0.1f0,0.0f0]
-tspan = (0.0f0,100.0f0)
-p = [1.9f0,0.1f0]
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqGPU&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# fix seeds&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.9f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleSummary&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fillalpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;time t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;X(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
-prob = SDEProblem(brusselator_f!,scalar_noise!,u0,tspan,p,noise=W)
 
-ensembleprob = EnsembleProblem(prob, prob_func = prob_func)
 
-sol = @time solve(ensembleprob,DRI1(),dt=0.1,EnsembleCPUArray(),trajectories=numtraj)
-sol_na = @time solve(ensembleprob,DRI1(),dt=0.8,adaptive=false,EnsembleCPUArray(),trajectories=numtraj)
 
 
-summ = EnsembleSummary(sol,0.0f0:0.5f0:100f0)
-pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quot;X(t)&amp;quot;, label= [&amp;quot;x₁(t)&amp;quot; &amp;quot;x₂(t)&amp;quot;], legend=true)
 
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -490,72 +412,55 @@ pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quo
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator_many_trajectories.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
+&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref3:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
 as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DifferentialEquations.EnsembleAnalysis
-meansol = timeseries_steps_mean(sol)
-meansol_na = timeseries_point_mean(sol_na,meansol.t)
-dts = []
-tmp1 = tspan[1]
-for tmp2 in meansol.t
-  global tmp1
-  push!(dts,tmp2-tmp1)
-  tmp1 = tmp2
-end
-#
-
-list_plots = []
-for i in 1:length(meansol.u)
-  l = @layout [a  b]
-  plt1 = plot(meansol[1, 1:i],meansol[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;adaptive&amp;quot;,
-         lw=2,
-         linecolor=1
-         )
-  plot!(meansol_na[1, 1:i],meansol_na[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;fixed step size&amp;quot;,
-         lw=2,
-         linecolor=2
-         )
-  pl2 = scatter(dts[1:i], xlabel = &amp;quot;step&amp;quot;, yaxis= &amp;quot;dtᵢ&amp;quot;, xlim = (0, length(meansol.u)),  ylim = (0.0, 2.3), legend=false)
-  plt = plot(plt1, pl2, layout = l)
-  push!(list_plots, plt)
-end
-
-anim = animate(list_plots,lw=2,every=1)
-
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleAnalysis&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_steps_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_point_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;global&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;#&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@layout&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;adaptive&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;fixed step size&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scatter&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;step&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;dtᵢ&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)),&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;2.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -567,53 +472,46 @@ anim = animate(list_plots,lw=2,every=1)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gpu-usage&#34;&gt;GPU usage&lt;/h3&gt;
 &lt;p&gt;All necessary tools to accelerate the simulation of (stochastic) differential equations on GPUs within the SciML
-ecosystem are collected in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
+ecosystem are collected in the &lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;Currently, bounds checking and return values are not allowed, i.e., functions must be
 written in the form:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f!(du,u,p,t)
-  @inbounds begin
-    du[1] = ..
-  end
-  nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;..&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
 Note that for some high weak order solvers GPU usage is not recommended as scalar indexing is used.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -622,21 +520,14 @@ Note that for some high weak order solvers GPU usage is not recommended as scala
       <link>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</link>
       <pubDate>Sat, 30 May 2020 15:10:33 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</guid>
-      <description>&lt;p&gt;First and foremost, I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
+      <description>&lt;p&gt;First and foremost, I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
 Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both  &lt;strong&gt;new high weak order solvers&lt;/strong&gt; as well as &lt;strong&gt;adjoint sensitivity methods&lt;/strong&gt; for stochastic differential equations (SDEs).
-The project is embedded within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
-Ultimately, the planned contributions will allow researchers to simulate (or even 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
+The project is embedded within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
+Ultimately, the planned contributions will allow researchers to simulate (or even &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
 Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.&lt;/p&gt;
 &lt;p&gt;This post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.&lt;/p&gt;
 &lt;h2 id=&#34;high-weak-order-solvers&#34;&gt;High Weak Order Solvers&lt;/h2&gt;
-&lt;p&gt;Currently, the 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
+&lt;p&gt;Currently, the &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
 In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.&lt;/p&gt;
 &lt;p&gt;However in many situations, we are only aiming for computing an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt;.
 In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods.
@@ -644,7 +535,7 @@ In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&
 As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.&lt;/p&gt;
 &lt;h3 id=&#34;second-order-runge-kutta-methods-for-ito-sdes&#34;&gt;Second order Runge-Kutta methods for Ito SDEs&lt;/h3&gt;
 &lt;p&gt;In the beginning of the community bonding period I finished the implementations of the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and &lt;code&gt;RI1()&lt;/code&gt;&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;. Specifically, only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn (in contrast to &lt;code&gt;m(m+1)/2&lt;/code&gt; from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.&lt;/p&gt;
-&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -653,47 +544,40 @@ $$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e7&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;. The code for a single  &lt;code&gt;dt&lt;/code&gt; reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-numtraj = 1e7
-u₀ = [1.0,1.0]
-function f!(du,u,p,t)
-  du[1] = -273//512*u[1]
-  du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-end
-function g!(du,u,p,t)
-  du[1,1] = 1//4*u[1]
-  du[1,2] = 1//16*u[1]
-  du[2,1] = (1-2*sqrt(2))/4*u[1]
-  du[2,2] = 1//10*u[1]+1//16*u[2]
-end
-dt = 1//8
-tspan = (0.0,10.0)
-prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-h(z) = z^2
-
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h(sol[end][1]),false)
-        )
-sol = solve(ensemble_prob, DRI1();
-        dt=dt,
-        save_start=false,
-        save_everystep=false,
-        weak_timeseries_errors=false,
-        weak_dense_errors=false,
-        trajectories=numtraj)
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
-&lt;pre&gt;&lt;code&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;();&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
 
 
-  
 
 
 
@@ -705,29 +589,24 @@ sol = solve(ensemble_prob, DRI1();
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/DRI1.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/DRI1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;DRI1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In the next couple of weeks, my focus will be on&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;adding other high weak order solvers,&lt;/li&gt;
 &lt;li&gt;implementing adaptive time stepping.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;More of our near-term goals are collected in this 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;More of our near-term goals are collected in this &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;adjoint-sensitivity-methods-for-sdes&#34;&gt;Adjoint Sensitivity Methods for SDEs&lt;/h2&gt;
-&lt;p&gt;In 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
+&lt;p&gt;In &lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
 &lt;p&gt;$$
 L = |f(p)-y|^2.
 $$&lt;/p&gt;
@@ -736,46 +615,43 @@ One possibility to obtain the gradient information for (stochastic) differential
 While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.&lt;/p&gt;
 &lt;p&gt;Alternatively to the &amp;ldquo;direct&amp;rdquo; AD approaches, the &lt;strong&gt;adjoint sensitivity method&lt;/strong&gt; can be used&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.
 Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).&lt;/p&gt;
-&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
+&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
 &lt;h3 id=&#34;reverse-sde-time-evolution&#34;&gt;Reverse SDE time evolution&lt;/h3&gt;
 &lt;p&gt;The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms.
 As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed.
 In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values.
-After some fixes for the 
-&lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
+After some fixes for the &lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
 &lt;p&gt;As an example, the code below computes first the forward evolution of an SDE&lt;/p&gt;
 &lt;p&gt;$$ dX  =  \alpha X dt + \beta X dW$$&lt;/p&gt;
 &lt;p&gt;with $\alpha=1.01$, $\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;  using StochasticDiffEq, DiffEqNoiseProcess
-
-  α=1.01
-  β=0.87
-
-  dt = 1e-3
-  tspan = (0.0,1.0)
-  u₀=1/2
-
-  tarray =  collect(tspan[1]:dt:tspan[2])
-
-  f!(du,u,p,t) = du .= α*u
-  g!(du,u,p,t) = du .= β*u
-
-
-  prob = SDEProblem(f!,g!,[u₀],tspan)
-  sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false)
-
-  _sol = deepcopy(sol) # to make sure the plot is correct
-  W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W))
-  prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1)
-  sol1 = solve(prob1,EulerHeun(),dt=dt)
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# to make sure the plot is correct&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NoiseGrid&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -787,18 +663,11 @@ After some fixes for the
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/animation.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/animation.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;animation.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gradients-of-diagonal-sdes&#34;&gt;Gradients of diagonal SDEs&lt;/h3&gt;
 &lt;p&gt;I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.&lt;/p&gt;
@@ -812,82 +681,80 @@ $$&lt;/p&gt;
 &lt;p&gt;acting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\&lt;br&gt;
+\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\
 \frac{d L}{d \beta}  &amp;amp;= 2 \sum_i W(t_i) |X(t_i)|^2
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Test, LinearAlgebra
-using DiffEqSensitivity, StochasticDiffEq
-using Random
-
-seed = 100
-Random.seed!(seed)
-
-u₀ = [0.5]
-tstart = 0.0
-tend = 0.1
-dt = 0.005
-trange = (tstart, tend)
-t = tstart:dt:tend
-tarray = collect(t)
-
-function g(u,p,t)
-  sum(u.^2.0)
-end
-
-function dg!(out,u,p,t,i)
-  (out.=-2.0*u)
-end
-
-p2 = [1.01,0.87]
-
-f(u,p,t) = p[1]*u
-σ(u,p,t) = p[2]*u
-
-
-Random.seed!(seed)
-prob = SDEProblem(f,σ,u₀,trange,p2)
-sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true)
-res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint())
-
-
-noise = vec((@. sol.W(tarray)))
-Wextracted = [W[1][1] for W in noise]
-resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp = [resp1, resp2]
-
-@test isapprox(res_p&#39;, resp, rtol = 1e-6)
-# True
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.005&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:tend&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.=-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RKMil&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;interpretation&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:Stratonovich&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res_u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adjoint_sensitivities&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vec&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;isapprox&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# True&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;finish the current backsolve adjoint version,&lt;/li&gt;
 &lt;li&gt;allow for computing the gradients of non-commuting SDEs,&lt;/li&gt;
 &lt;li&gt;implement also an interpolation adjoint version,&lt;/li&gt;
 &lt;li&gt;benchmark it with respect to AD approaches&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
diff --git a/tag/gsoc-2020/page/1/index.html b/tag/gsoc-2020/page/1/index.html
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+
   
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-    <div>
-      <h2><a href="/post/gsoc-2021/" >Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis</a></h2>
-      <div class="article-style">
-        
-          Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.
-        
-      </div>
-    </div>
   
     
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-    
-    <div>
-      <h2><a href="/post/abstract_differentiation/" >AbstractDifferentiation.jl for AD-backend agnostic code </a></h2>
-      <div class="article-style">
-        
-          Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris' article3 for a detailed examination.
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+      <a href="/post/gsoc-2021/" >Neural Hybrid Differential Equations and Adjoint Sensitivity Analysis</a>
+    </div>
+
     
-    
-    
-    <div>
-      <h2><a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a></h2>
+    <a href="/post/gsoc-2021/"  class="summary-link">
       <div class="article-style">
-        
-          In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
-        
+        Project summary In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.
       </div>
-    </div>
-  
-    
-    
+    </a>
     
-    <div>
-      <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></h2>
-      <div class="article-style">
-        
-          In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
+
+    <div class="stream-meta article-metadata">
+
+      
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+      
         
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+      
     
-    <div>
-      <h2><a href="/post/hybridde/" >Neural Hybrid Differential Equations</a></h2>
-      <div class="article-style">
-        
-          I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
-        
-      </div>
-    </div>
+    Aug 13, 2021
+  </span>
   
 
   
 
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+  <span class="middot-divider"></span>
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+    3 min read
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+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/abstract_differentiation/" >AbstractDifferentiation.jl for AD-backend agnostic code </a>
+    </div>
+
     
-    <script>const code_highlighting = true;</script>
+    <a href="/post/abstract_differentiation/"  class="summary-link">
+      <div class="article-style">
+        Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris&rsquo; article3 for a detailed examination.
+      </div>
+    </a>
     
 
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+    Aug 1, 2021
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+    11 min read
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+      <a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a>
+    </div>
+
     
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+    <a href="/post/bouncing_ball/"  class="summary-link">
+      <div class="article-style">
+        In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
+      </div>
+    </a>
     
 
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+      Frank Schäfer and Moritz Schauer</span>
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+      
     
+    Jul 16, 2021
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+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
+    </div>
 
     
+    <a href="/post/shadowing/"  class="summary-link">
+      <div class="article-style">
+        In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
+      </div>
+    </a>
     
 
+    <div class="stream-meta article-metadata">
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+    Jul 2, 2021
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+    9 min read
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+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/hybridde/" >Neural Hybrid Differential Equations</a>
+    </div>
+
     
+    <a href="/post/hybridde/"  class="summary-link">
+      <div class="article-style">
+        I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
+      </div>
+    </a>
     
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+    <div class="stream-meta article-metadata">
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+    Jun 16, 2021
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+  <span class="article-reading-time">
+    7 min read
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@@ -734,32 +1237,185 @@ <h2><a href="/post/hybridde/" >Neural Hybrid Differential Equations</a></h2>
 
   
   
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         </button>
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@@ -786,5 +1443,25 @@ <h5 class="modal-title">Cite</h5>
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diff --git a/tag/gsoc-2021/index.xml b/tag/gsoc-2021/index.xml
index 7d0b32a..88f6dda 100644
--- a/tag/gsoc-2021/index.xml
+++ b/tag/gsoc-2021/index.xml
@@ -5,9 +5,9 @@
     <link>https://frankschae.github.io/tag/gsoc-2021/</link>
       <atom:link href="https://frankschae.github.io/tag/gsoc-2021/index.xml" rel="self" type="application/rss+xml" />
     <description>GSoC 2021</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Fri, 13 Aug 2021 21:41:45 +0200</lastBuildDate>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Fri, 13 Aug 2021 21:41:45 +0200</lastBuildDate>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
       <title>GSoC 2021</title>
       <link>https://frankschae.github.io/tag/gsoc-2021/</link>
     </image>
@@ -19,128 +19,91 @@
       <guid>https://frankschae.github.io/post/gsoc-2021/</guid>
       <description>&lt;h2 id=&#34;project-summary&#34;&gt;Project summary&lt;/h2&gt;
 &lt;p&gt;In this project, we have implemented state-of-the-art sensitivity tools for chaotic dynamical systems, continuous adjoint sensitivity methods for hybrid differential equations, as well as a high level API for automatic differentiation.&lt;/p&gt;
-&lt;p&gt;Possible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/projects/#5357798591823872&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the GSoC page&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Possible fields of application for these tools range from model discovery with explicit dosing times in pharmacology, over accurate gradient estimates for chaotic fluid dynamics, to the control of open quantum systems. A more detailed summary is available &lt;a href=&#34;https://summerofcode.withgoogle.com/projects/#5357798591823872&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the GSoC page&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;blog-posts&#34;&gt;Blog posts&lt;/h2&gt;
 &lt;p&gt;The following blog posts describe the work throughout the GSoC period in more detail:&lt;/p&gt;
 &lt;ol&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/hybridde/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Neural Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/shadowing/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/bouncing_ball/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity Analysis of Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://frankschae.github.io/post/abstract_differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl for AD-backend agnostic code&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/hybridde/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Neural Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/shadowing/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/bouncing_ball/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity Analysis of Hybrid Differential Equations&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://frankschae.github.io/post/abstract_differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl for AD-backend agnostic code&lt;/a&gt;&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;docs&#34;&gt;Docs&lt;/h2&gt;
-&lt;p&gt;Documentation with respect to the adjoint sensitivity tools will be available 
-&lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the local sensitivity analysis&lt;/a&gt; and 
-&lt;a href=&#34;http://scimlbase.sciml.ai/dev/fundamentals/Differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the control of automatic differentiation choices&lt;/a&gt; pages.&lt;/p&gt;
+&lt;p&gt;Documentation with respect to the adjoint sensitivity tools will be available &lt;a href=&#34;https://diffeq.sciml.ai/latest/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the local sensitivity analysis&lt;/a&gt; and &lt;a href=&#34;http://scimlbase.sciml.ai/dev/fundamentals/Differentiation/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;on the control of automatic differentiation choices&lt;/a&gt; pages.&lt;/p&gt;
 &lt;h2 id=&#34;achievements&#34;&gt;Achievements&lt;/h2&gt;
 &lt;p&gt;Below is a list of PRs in the various repositories in chronological order.&lt;/p&gt;
 &lt;h4 id=&#34;diffeqsensitivityjl&#34;&gt;DiffEqSensitivity.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/415&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add additive noise downstream test for DiffEqFlux&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/416&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallback fixes&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/417&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for changes of p in callbacks&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/418&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix for using the correct uleft/pleft in continuous callback&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/419&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix broadcasting error on steady state adjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/420&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Forward Least Squares Shadowing (LSS)&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/422&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint-mode for the LSS method&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/423&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve dispatch for LSS methods&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/437&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Non-Intrusive Least Square Shadowing (NILSS)&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/442&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve for NILSS&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/443&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove allocation in NILSS&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/444&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Handle additional callback case&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;State-dependent Continuous Callbacks for BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/474&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;QuadratureAdjoint() for ContinuousCallback&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/475&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;More tests for Neural ODEs with callbacks for different sensitivity algorithms&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/476&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support for PeriodicCallbacks in continuous adjoint methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/415&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add additive noise downstream test for DiffEqFlux&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/416&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallback fixes&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/417&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow for changes of p in callbacks&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/418&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix for using the correct uleft/pleft in continuous callback&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/419&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix broadcasting error on steady state adjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/420&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Forward Least Squares Shadowing (LSS)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/422&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Adjoint-mode for the LSS method&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/423&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve dispatch for LSS methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/437&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Non-Intrusive Least Square Shadowing (NILSS)&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/442&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;concrete_solve for NILSS&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/443&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove allocation in NILSS&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/444&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Handle additional callback case&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;State-dependent Continuous Callbacks for BacksolveAdjoint&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/474&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;QuadratureAdjoint() for ContinuousCallback&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/475&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;More tests for Neural ODEs with callbacks for different sensitivity algorithms&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/476&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Support for PeriodicCallbacks in continuous adjoint methods&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;abstractdifferentiationjl&#34;&gt;AbstractDifferentiation.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/2&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes gradient, Jacobian, Hessian, and vjp tests&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/2&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fixes gradient, Jacobian, Hessian, and vjp tests&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;Open:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/3&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add ForwardDiff and Zygote&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/3&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add ForwardDiff and Zygote&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;ordinarydiffeqjl&#34;&gt;OrdinaryDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/OrdinaryDiffEq.jl/pull/1424&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix discrete reverse mode for some standard controllers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/OrdinaryDiffEq.jl/pull/1424&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fix discrete reverse mode for some standard controllers&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqcallbacksjl&#34;&gt;DiffEqCallbacks.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqCallbacks.jl/pull/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Introduce a PeriodicCallbackAffect struct&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqCallbacks.jl/pull/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Introduce a PeriodicCallbackAffect struct&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;steadystatediffeqjl&#34;&gt;SteadyStateDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/SteadyStateDiffEq.jl/pull/31&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Convert alg.tspan to type of prob.u0&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/SteadyStateDiffEq.jl/pull/31&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Convert alg.tspan to type of prob.u0&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;chainrulesjl&#34;&gt;ChainRules.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/506&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Do not differentiate through the construction of BitArray&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/508&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Use splatting in BitArray&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/506&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Do not differentiate through the construction of BitArray&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl/pull/508&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Use splatting in BitArray&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqnoiseprocessjl&#34;&gt;DiffEqNoiseProcess.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/94&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow solvers to use Noise Grid with SVectors&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqNoiseProcess.jl/pull/94&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Allow solvers to use Noise Grid with SVectors&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;stochasticdiffeqjl&#34;&gt;StochasticDiffEq.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/428&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove Ihat2 matrix from weak solvers&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/pull/428&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Remove Ihat2 matrix from weak solvers&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h4 id=&#34;diffeqdocsjl&#34;&gt;DiffEqDocs.jl&lt;/h4&gt;
 &lt;p&gt;Merged:&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/490&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Small typo on plot page&lt;/a&gt;&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/492&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add docs for shadowing methods&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/490&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Small typo on plot page&lt;/a&gt;&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://github.com/SciML/DiffEqDocs.jl/pull/492&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Add docs for shadowing methods&lt;/a&gt;&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;future-work&#34;&gt;Future work&lt;/h2&gt;
 &lt;p&gt;Besides the implementation of more shadowing methods, such as&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/1801.08674&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NILSAS&lt;/a&gt;,&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/1711.06633&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FD-NILSS&lt;/a&gt;, or&lt;/li&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://arxiv.org/abs/2009.00595&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fast linear response&lt;/a&gt;,&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/1801.08674&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NILSAS&lt;/a&gt;,&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/1711.06633&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FD-NILSS&lt;/a&gt;, or&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://arxiv.org/abs/2009.00595&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Fast linear response&lt;/a&gt;,&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;we are planning to&lt;/p&gt;
 &lt;ul&gt;
@@ -148,20 +111,10 @@
 &lt;li&gt;refine the AbstractDifferentiation.jl package and use it within DiffEqSensitivity.jl,&lt;/li&gt;
 &lt;li&gt;add more docs and examples.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the 
-&lt;a href=&#34;https://discourse.julialang.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julia language discourse&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;If you have any further suggestions or comments, check out our slac/zulip channels #sciml-bridged and #diffeq-bridged or the &lt;a href=&#34;https://discourse.julialang.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julia language discourse&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;acknowledgement&#34;&gt;Acknowledgement&lt;/h2&gt;
-&lt;p&gt;Many thanks to my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times.
-I would also like to thank 
-&lt;a href=&#34;https://quantumtheory-bruder.physik.unibas.ch/en/people/group-members/christoph-bruder/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/arnoldjulian&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julian Arnold&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mako-git&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Martin Koppenhöfer&lt;/a&gt; for helpful comments on my blog posts. Special thanks to 
-&lt;a href=&#34;https://github.com/Zymrael&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Michael Poli&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/massastrello&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stefano Massaroli&lt;/a&gt; for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
+&lt;p&gt;Many thanks to my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; for their unique, continuous support. It was a great opportunity to be part of such an inspiring collaboration. I highly appreciate our quick and flexible meeting times.
+I would also like to thank &lt;a href=&#34;https://quantumtheory-bruder.physik.unibas.ch/en/people/group-members/christoph-bruder/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Christoph Bruder&lt;/a&gt;, &lt;a href=&#34;https://github.com/arnoldjulian&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Julian Arnold&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mako-git&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Martin Koppenhöfer&lt;/a&gt; for helpful comments on my blog posts. Special thanks to &lt;a href=&#34;https://github.com/Zymrael&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Michael Poli&lt;/a&gt; and &lt;a href=&#34;https://github.com/massastrello&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Stefano Massaroli&lt;/a&gt; for their suggestions on adjoints for hybrid differential equations. Finally, thanks to the very supportive julia community and to Google&amp;rsquo;s open source program for funding this experience!&lt;/p&gt;
 </description>
     </item>
     
@@ -170,33 +123,17 @@ I would also like to thank
       <link>https://frankschae.github.io/post/abstract_differentiation/</link>
       <pubDate>Sun, 01 Aug 2021 12:03:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/abstract_differentiation/</guid>
-      <description>&lt;p&gt;
-&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is 
-&lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&#39; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and 
-&lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, 
-&lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different 
-&lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by 
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a 
-&lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
-&lt;p&gt;
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
+      <description>&lt;p&gt;&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is &lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&amp;rsquo; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and &lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, &lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different &lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by &lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a &lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
+&lt;p&gt;&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
 &lt;h2 id=&#34;optimization-of-the-rosenbrock-function&#34;&gt;Optimization of the Rosenbrock function&lt;/h2&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
+&lt;p&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
 &lt;p&gt;$$
 g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2.
 $$&lt;/p&gt;
 &lt;p&gt;The function $g$ has a global minimum at $(x_1^\star, x_2^\star)= (a, a^2)$ with $g(x_1^\star, x_2^\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.&lt;/p&gt;
-&lt;p&gt;Let us now implement the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
+&lt;p&gt;Let us now implement the &lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
 &lt;p&gt;$$
 S({\bf x}) = \frac{1}{2} \sum_{i=1}^M f_i({\bf x})^2.
 $$&lt;/p&gt;
@@ -204,55 +141,57 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 {\bf x}^{k+1} = {\bf x}^k - \alpha_k \left(J^T J \right)^{-1} J^T f({\bf x}^k),
 $$
-where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a 
-&lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
+where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a &lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;The following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\bf x^\star}=(1,1)$ and our initial guess ${\bf x_0}=(0,-0.1)$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Pkg
-path = @__DIR__
-cd(path); Pkg.activate(&amp;quot;.&amp;quot;); Pkg.instantiate()
-
-## AbstractDifferentiation is not released yet!!
-
-using AbstractDifferentiation
-using Test, LinearAlgebra
-using FiniteDifferences, ForwardDiff, Zygote
-using Enzyme, UnPack
-using Plots, LaTeXStrings
-# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6
-
-## Rosenbrock function
-# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²
-g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@__DIR__&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;activate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;.&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;instantiate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## AbstractDifferentiation is not released yet!!&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AbstractDifferentiation&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Enzyme&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;UnPack&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Rosenbrock function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualization&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt;    
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.6&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.5&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;linealpha&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;colorbar&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;camera&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;heatmap&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_0&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:circle&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:green&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x^\star&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:star&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:red&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Rosenbrock.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-# visualization
-p = [1.0,100.0]
-x₀ = [0.0,-0.1]
-xopt = [1.0,1.0]
 
-do_plot = true
-if do_plot    
-    x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5
-    z = Surface((x₁,x₂)-&amp;gt;g([x₁,x₂],p), x₁, x₂)
-    pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true,
-                labelfontsize=20,camera = (3,50),
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
 
-    pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp),
-                labelfontsize=20,
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
-    scatter!(pl2, [(x₀[1],x₀[2])], label=L&amp;quot;x_0&amp;quot;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green)
-    scatter!(pl2, [(xopt[1],xopt[2])],label=L&amp;quot;x^\star&amp;quot;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red)
 
-    pl = plot(pl1,pl2, layout=(2,1))
-    savefig(pl, &amp;quot;Rosenbrock.png&amp;quot;)
-end
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -261,241 +200,216 @@ end
 
 
 
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Rosenbrock.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;To apply the Gauss-Newton algorithm to the Rosenbrock function $g$, we first cast $g$ into an appropriate form fulfilling $S({\bf x})$, i.e., we use:&lt;/p&gt;
 &lt;p&gt;$$
 f:\mathbb{R}^2\rightarrow\mathbb{R}^2:  {\bf x} \mapsto \begin{pmatrix}
-f_1({\bf x}) \\&lt;br&gt;
-f_2({\bf x}) \\&lt;br&gt;
+f_1({\bf x}) \\
+f_2({\bf x}) \\
 \end{pmatrix} = \begin{pmatrix}
-\sqrt{2}(a-x_1) \\&lt;br&gt;
-\sqrt{2b}(x_2-x_1^2)\\&lt;br&gt;
+\sqrt{2}(a-x_1) \\
+\sqrt{2b}(x_2-x_1^2)\\
 \end{pmatrix},
 $$&lt;/p&gt;
 &lt;p&gt;instead of $g$. We can easily compute the Jacobian of $f$ manually&lt;/p&gt;
 &lt;p&gt;$$
 J =  \begin{pmatrix}
--\sqrt{2} &amp;amp; 0 \\&lt;br&gt;
--2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\&lt;br&gt;
+-\sqrt{2} &amp;amp; 0 \\
+-2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\
 \end{pmatrix}.
 $$&lt;/p&gt;
 &lt;p&gt;We can then implement a (simple, non-optimized) version of the Gauss-Newton algorithm as follows.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# bring Rosenbrock function into the form &amp;quot;sum of squares of functions&amp;quot;
-f1(x,p) = convert(eltype(x),sqrt(2))*(p[1]-x[1])
-f2(x,p) = convert(eltype(x),sqrt(2*p[2]))*(x[2]-x[1]^2)
-f(x,p) = [f1(x,p),f2(x,p)]
-function f(res,x,p) # Enzyme works with inplace functions
-	res[1] = f1(x,p)
-	res[2] = f2(x,p)
-	return nothing
-end
-
-## manually pre-defined Jacobian
-function Jacobian(x,p)
-  [-convert(eltype(x),sqrt(2))   0
-  -2*x[1]*convert(eltype(x),sqrt(2*p[2]))  convert(eltype(x),sqrt(2*p[2]))]
-end
-
-## Gauss-Newton scheme
-function GaussNewton!(xs, x, p; maxiter=8, backend=nothing)
-    for i=1:maxiter
-        x = step(x, p, backend)
-        @info i
-        @show x
-        push!(xs, x)
-    end
-    return xs, x
-end
-done(x,x2,p) = g(x2,p) &amp;lt; g(x,p)
-function step(x, p, backend::Nothing, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = Jacobian(x,p)
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;xs = [x₀]
-GaussNewton!(xs, x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# output:
-[ Info: 1 ]
-x = [0.125, -0.08750000000000001]
-[ Info: 2 ]
-x = [0.234375, -0.047265625000000006]
-[ Info: 3 ]
-x = [0.4257812499999995, 0.06800537109374968]
-[ Info: 4 ]
-x = [0.5693359374999986, 0.21857223510742047]
-[ Info: 5 ]
-x = [0.784667968749996, 0.5165503501892037]
-[ Info: 6 ]
-x = [0.9999999999999961, 0.9536321163177449]
-[ Info: 7 ]
-x = [0.9999999999999989, 0.9999999999999999]
-[ Info: 8 ]
-x = [1.0, 1.0]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of 
-&lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## FiniteDifferences
-struct FDMBackend{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend() = FDMBackend(central_fdm(5, 1))
-const fdm_backend = FDMBackend()
-# Minimal interface
-AD.@primitive function jacobian(ab::FDMBackend, f, xs...)
-    return FiniteDifferences.jacobian(ab.alg, f, xs...)
-end
-
-# AD Jacobian returns tuple
-# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]
-# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]
-
-@test AD.jacobian(fdm_backend, x-&amp;gt;f(x,p), x₀)[1] ≈ Jacobian(x₀, p)
-@test AD.jacobian(fdm_backend, f, x₀, p)[1] ≈ Jacobian(x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function step(x, p, backend, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = AD.jacobian(backend, f, x, p)[1]
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-
-
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=fdm_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If we want to use reverse-mode AD instead, for example via 
-&lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Zygote
-struct ZygoteBackend &amp;lt;: AD.AbstractReverseMode end
-const zygote_backend = ZygoteBackend()
-AD.@primitive function pullback_function(ab::ZygoteBackend, f, xs...)
-    return function (vs)
-        # Supports only single output
-        _, back = Zygote.pullback(f, xs...)
-        if vs isa AbstractVector
-            back(vs)
-        else
-            @assert length(vs) == 1
-            back(vs[1])
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(zygote_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=zygote_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (
-&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## ForwardDiff
-struct ForwardDiffBackend &amp;lt;: AD.AbstractForwardMode end
-const forwarddiff_backend = ForwardDiffBackend()
-AD.@primitive function pushforward_function(ab::ForwardDiffBackend, f, xs...)
-    # jvp = f&#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0
-    return function (vs)
-        if xs isa Tuple
-            @assert length(xs) &amp;lt;= 2
-            if length(xs) == 1
-                (ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1]),0),)
-            else
-                ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1], xs[2]+h*vs[2]),0)
-            end
-        else
-            ForwardDiff.derivative(h-&amp;gt;f(xs+h*vs),0)
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(forwarddiff_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=forwarddiff_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;where we have used that the Jacobian-vector product $f&#39;(x)v$, i.e., the primitives of forward-mode AD, can be computed by 
-&lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. 
-&lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Enzyme
-struct EnzymeBackend{T1,T2,T3,T4} &amp;lt;: AD.AbstractReverseMode
-    out::T1
-    λ::T2
-    ∂f_∂x::T3
-    ∂f_∂p::T4
-end
-
-out = zero(x₀)
-λ = zero(x₀)
-∂f_∂x = zero(x₀)
-∂f_∂p = zero(p)
-
-const enzyme_backend = EnzymeBackend(out,λ,∂f_∂x,∂f_∂p)
-AD.@primitive function pullback_function(ab::EnzymeBackend, f, xs...)
-    return function (vs)  
-        # enzyme works only with inplace functions
-        if !(vs isa AbstractVector)
-            @assert length(vs) == 1 # Supports only single output
-            vs = vs[1]
-        end
-
-        if xs isa Tuple
-            @assert length(xs) == 2  # hard-coded for use case with two inputs
-            x₀ = xs[1]
-            p = xs[2]
-        end
-
-        @unpack out, λ, ∂f_∂x, ∂f_∂p = ab # cached in the struct, could also be created in here
-
-        ∂f_∂x .*= false
-        ∂f_∂p .*= false
-        out .*= false
-
-        copyto!(λ, vs)
-
-        autodiff(Duplicated(out, λ), Duplicated(x₀, ∂f_∂x), Duplicated(p, ∂f_∂p)) do _out,_x, _p
-            f(_out,_x,_p)
-        end
-        return (∂f_∂x,∂f_∂p)
-    end
-end
-AD.isinplace(ab::EnzymeBackend) = true
-AD.primalvalue(ab::EnzymeBackend, nothing, f, xs) = (f(ab.out,xs...);return ab.out)
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(enzyme_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=enzyme_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# bring Rosenbrock function into the form &amp;#34;sum of squares of functions&amp;#34;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Enzyme works with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## manually pre-defined Jacobian&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;   &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Gauss-Newton scheme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# output:&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.125&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.08750000000000001&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.234375&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.047265625000000006&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.4257812499999995&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.06800537109374968&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5693359374999986&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.21857223510742047&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.784667968749996&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5165503501892037&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;6&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999961&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9536321163177449&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;7&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999989&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9999999999999999&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of &lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## FiniteDifferences&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Minimal interface&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# AD Jacobian returns tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If we want to use reverse-mode AD instead, for example via &lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## ForwardDiff&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractForwardMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c&#34;&gt;# jvp = f&amp;#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;where we have used that the Jacobian-vector product $f&amp;rsquo;(x)v$, i.e., the primitives of forward-mode AD, can be computed by &lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. &lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Enzyme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;  
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# enzyme works only with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;  &lt;span class=&#34;c&#34;&gt;# hard-coded for use case with two inputs&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@unpack&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# cached in the struct, could also be created in here&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;autodiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;do&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;isinplace&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;primalvalue&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
 &lt;h2 id=&#34;some-current-glitches&#34;&gt;Some current glitches&lt;/h2&gt;
 &lt;p&gt;First, the push forward of a tuple of vectors, e.g., $(v_1, v_2)$, for a function with several input arguments is currently ambiguous. While &lt;code&gt;AD.jacobian&lt;/code&gt; primitives and &lt;code&gt;AD.pullback_function&lt;/code&gt; primitives interpret the push forward of our $f$ function as&lt;/p&gt;
 &lt;p&gt;$$
@@ -505,86 +419,77 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\partial f(x_0,p)}{\partial x} v_1 + \frac{\partial f(x_0,p)}{\partial p} v_2.
 $$&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous
-vs = (randn(2), randn(2))
-res1 = AD.pushforward_function(fdm_backend, f, x₀, p)(vs)
-res2 = AD.pushforward_function(forwarddiff_backend, f, x₀, p)(vs)
-
-@test res2 ≈ res1[1] + res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;res2a = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((vs[1], zero(vs[2])))
-res2b = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((zero(vs[1]), vs[2]))
-
-@test res2a ≈ res1[1]
-@test res2b ≈ res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;struct FDMBackend2{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend2() = FDMBackend2(central_fdm(5, 1))
-const fdm_backend2 = FDMBackend2()
-AD.@primitive function pushforward_function(ab::FDMBackend2, f, xs...)
-    return function (vs)
-        FDM.jvp(ab.alg, f, tuple.(xs, vs)...)
-    end
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian only defined with respect to single input variable
-@test_throws AssertionError H1 = AD.hessian(forwarddiff_backend, g, x₀, p)
-H1 = AD.hessian(forwarddiff_backend, x-&amp;gt;g(x,p), x₀)
-H2 = AD.hessian(forwarddiff_backend, p-&amp;gt;g(x₀,p), p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)
-backends = AD.HigherOrderBackend((forwarddiff_backend,zygote_backend))
-H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;FDM&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jvp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tuple&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian only defined with respect to single input variable&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test_throws&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AssertionError&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;HigherOrderBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;There are many other use cases, e.g.,&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;AbstractDifferentiation.jl is by no means complete yet. We are still in the very early stages, but we hope to make significant progress in the coming weeks. Some of the next steps are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;fixing remaining bugs, e.g., with respect to the computation of the Hessian and&lt;/li&gt;
-&lt;li&gt;adding AD/Finite Differentiation packages such as 
-&lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
+&lt;li&gt;adding AD/Finite Differentiation packages such as &lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -596,69 +501,77 @@ H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
       <description>&lt;p&gt;In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. We highlight differences between explicit&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and implicit events&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;. As a paradigmatic example, we consider a bouncing ball described by the ODE&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\&lt;br&gt;
+\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\
 \text{d}v(t) &amp;amp;= -\mathrm g\thinspace  \text{d}t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;with initial condition&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t=0) &amp;amp;= z_0 = 5, \\&lt;br&gt;
+z(t=0) &amp;amp;= z_0 = 5, \\
 v(t=0) &amp;amp;= v_0 = -0.1.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;The initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.&lt;/p&gt;
-&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
+&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
 &lt;p&gt;When ignoring the bounces, we can straightforwardly integrate the ODE analytically&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\&lt;br&gt;
+z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\
 v(t) &amp;amp;= v_0 - \mathrm g\thinspace  t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;or numerically using the OrdinaryDiffEq package from the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### simulate forward
+&lt;p&gt;or numerically using the OrdinaryDiffEq package from the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### simulate forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# dynamics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# parameters and solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;5.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.9&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward_no_bounce.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
+
+
 
-using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity
-using Plots, LaTeXStrings
 
-# dynamics
-function f(du,u,p,t)
-  du[1] = u[2]
-  du[2] = -p[1]
-end
 
-# parameters and solve
-z0 = 5.0
-v0 = -0.1
-t0 = 0.0
-tend = 1.9
-g = 10
-γ = 0.8
 
-u0 = [z0,v0]
-tspan = (t0,tend)
-p = [g, γ]
-prob = ODEProblem(f,u0,tspan,p)
 
-# plot forward trajectory
-sol = solve(prob,Tsit5(),saveat=0.1)
-pl = plot(sol, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomleft)
-hline!(pl, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34;&gt;&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
+&lt;/p&gt;
 
 
 
@@ -667,25 +580,27 @@ savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+
+
+
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward_no_bounce.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Of course, this way the ball continues to fall through the barrier at $z=0$.&lt;/p&gt;
 &lt;h2 id=&#34;forward-simulation-with-events&#34;&gt;Forward simulation with events&lt;/h2&gt;
 &lt;p&gt;At time $\tau$ around $\tau \approx 1$, the ball hits the ground $z(\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\tau$ with new initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau}) &amp;amp;=  z(\tau-) ,\\&lt;br&gt;
+z({\tau}) &amp;amp;=  z(\tau-) ,\\
 v({\tau})&amp;amp;= -\gamma v(\tau-) ,
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -702,15 +617,15 @@ $$&lt;/p&gt;
 &lt;p&gt;The full explicit trajectory $z_{\rm exp}(t) = z(t)$ is determined by&lt;/p&gt;
 &lt;p&gt;$$
 z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\&lt;br&gt;
--0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\
+-0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\
 \quad  - (-0.99005 + t) (-0.99005 \mathrm g + v_0)\gamma  ,&amp;amp; \forall t \ge \tau,
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;where we used&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\&lt;br&gt;
+z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\
 v({\tau})&amp;amp;= -\gamma v({\tau-}) = -\gamma(v_0 - 0.99005 \mathrm g)  .
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -719,15 +634,14 @@ $$&lt;/p&gt;
 a(z,v) = (z, -\gamma v).
 $$&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;DiscreteCallback&lt;/code&gt; in this case to simulate the system.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with DiscreteCallback (explicit event)
-tstar = (v0 + sqrt(v0^2+2*z0*g))/g
-condition1(u,t,integrator) = (t == tstar)
-affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2]
-cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true))
-
-sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar])
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with DiscreteCallback (explicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiscreteCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstops&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
 &lt;h3 id=&#34;implicit-events&#34;&gt;Implicit events&lt;/h3&gt;
 &lt;p&gt;The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)&lt;/p&gt;
 &lt;p&gt;$$
@@ -736,7 +650,7 @@ $$&lt;/p&gt;
 &lt;p&gt;where an event occurs at time $\tau$ if $g(z(\tau),v(\tau),p,\tau) = 0$. We have already used this condition to define our impact time $\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time  $g(z,v,p,t) = 0$.&lt;/p&gt;
 &lt;p&gt;As in the previous case, we can analytically compute the full trajectory of the ball. By substituting the formula for $\tau$ we have at the event time&lt;/p&gt;
 &lt;p&gt;\begin{aligned}
-z({\tau})&amp;amp;= 0, \\&lt;br&gt;
+z({\tau})&amp;amp;= 0, \\
 v({\tau}-)&amp;amp;= - \sqrt{v_0^2 + 2 \mathrm g z_0}
 \end{aligned}&lt;/p&gt;
 &lt;p&gt;for the left limit and&lt;/p&gt;
@@ -746,33 +660,30 @@ v({\tau})&amp;amp;= \gamma \sqrt{v_0^2 + 2 \mathrm g z_0}
 &lt;p&gt;right after the bounce. Thus, the full trajectory $z_{\rm imp}(t) = z(t)$ is given by&lt;/p&gt;
 &lt;p&gt;$$
 (\star) \quad z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\&lt;br&gt;
--\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\
+-\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\
 \quad\cdot \space (-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0} (1 + 2 \gamma)), &amp;amp; \forall t \ge \tau.
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;This is correct even if one substitutes, e.g., a value with higher precision $\mathrm g = 9.81$ for the gravitation constant.&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;ContinuousCallback&lt;/code&gt; in this case.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with ContinuousCallback (implicit event)
-condition2(u,t,integrator) = u[1] # Event happens when condition2(u,t,integrator) == 0
-cb2 = ContinuousCallback(condition2,affect!,save_positions=(true,true))
-sol2 = solve(prob,Tsit5(),callback=cb2,saveat=0.1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# plot forward trajectory
-pl1 = plot(sol1, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;explicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-pl2 = plot(sol2, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;implicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-hline!(pl1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-hline!(pl2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-pl = plot(pl1,pl2)
-savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with ContinuousCallback (implicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Event happens when condition2(u,t,integrator) == 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ContinuousCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;explicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;implicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
-  
 
 
 
@@ -784,18 +695,12 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In addition, the implicitly defined impact time via the &lt;code&gt;ContinuousCallback&lt;/code&gt; also changes appropriately when changing the initial conditions or the parameters, for example when using  $\mathrm g = 9.81$ for the gravitation constant. In other words, the event time $\tau=\tau(p,z_0,v_0,t_0)$ is a function of the parameters and initial conditions, and is implicitly defined by the event condition.&lt;/p&gt;
 &lt;p&gt;Suppose we let the ball drop from a somewhat higher position now. Does an increase in height $z$ at $t=0$ give an increase or decrease in height at the end time $t_\text{end}=1.9$? This is something we can answer with sensitivity analysis. For example if we increase the height by (a fraction of) one unit then using $(\star)$&lt;/p&gt;
@@ -804,36 +709,36 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 $$&lt;/p&gt;
 &lt;p&gt;meaning the height at $t_\text{end}$ is also by a corresponding fraction of 0.84 units higher.&lt;/p&gt;
 &lt;p&gt;We can verify this visually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# animate forward trajectory
-sol3 = solve(remake(prob,u0=[u0[1]+0.5,u0[2]]),Tsit5(),callback=cb2,saveat=0.01)
-
-plt2 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;, xlims=(t0,tend))
-hline!(plt2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-plot!(plt2, sol3, tspan=(t0,tend), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)
-
-list_plots = []
-for t in sol3.t
-  tstart = 0.0
-
-  plt1 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;)
-  hline!(plt1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-  plot!(plt1, sol3, tspan=(t0,t), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-  scatter!(plt1,[t,t], sol3(t), color=[1, 2], label=false)
-  plt = plot(plt1,plt2)
-  push!(list_plots, plt)
-end
-
-plot(list_plots[100])
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# animate forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]]),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-anim = animate(list_plots,every=1)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -845,18 +750,10 @@ anim = animate(list_plots,every=1)
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/BB.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;The original curve is shown in black in the figure above.&lt;/p&gt;
 &lt;h2 id=&#34;sensitivity-analysis-with-events&#34;&gt;Sensitivity analysis with events&lt;/h2&gt;
@@ -871,7 +768,7 @@ $$&lt;/p&gt;
 &lt;p&gt;Inserting our results for $z(t) = z_{\rm exp}(t)$ instead of $z(t) = z_{\rm imp}(t)$ at $t_{\text{end}}$, a different value for the sensitivity is obtained (a value ignoring the changes in $z$ due to changes in the bouncing time $\tau$), e.g.,&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L^{\text{exp}}}{\text{d} z_0} = \begin{cases}
-1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\&lt;br&gt;
+1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\
 0.84 &amp;amp; \text{with } z(t) = z_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -882,7 +779,7 @@ $$&lt;/p&gt;
 &lt;p&gt;we could be interested in the sensitivity of $L$ with respect $g$. Using the velocity $v(t)$ at the explicit time $t = 0.9905$ instead of the implicit $v(\tau-)$, again a different value for the sensitivity is obtained (a value again ignoring the changes in $\tau$ and not the one we are looking for), even though $\tau = 0.9905$ in both cases:&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L}{\text{d} g} = \begin{cases}
--0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\&lt;br&gt;
+-0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\
 -\frac{g}{\sqrt{v_0^2 + 2 g z_0}} = -0.99995 &amp;amp; \text{ with } v(t) = v_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -899,51 +796,48 @@ $$&lt;/p&gt;
 &lt;p&gt;We write&lt;/p&gt;
 &lt;p&gt;$$\text{solve}(t_0, x_0, t, p)$$&lt;/p&gt;
 &lt;p&gt;$(= x(t))$ for the functional solution of the ODE at time $t$.&lt;/p&gt;
-&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
+&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
 &lt;p&gt;Let us focus on the &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; algorithm which computes the sensitivities&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\&lt;br&gt;
+\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\
 \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}p} &amp;amp;= \lambda_{p}(t_{0}),
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;of a loss function $L$ acting on the final state with respect to the initial state and the parameters. It does so by solving an ODE for $\lambda(s)$ in reverse time from $t$ to $t_0$&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\&lt;br&gt;
+\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\
 \frac{\text{d}\lambda_{p}(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(x(s), \rightarrow p, s)}{\text{d} p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;with initial conditions:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\&lt;br&gt;
+\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\
 \lambda_{p}(t) &amp;amp;= 0.
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The arrows ($\rightarrow$) indicate the variable with respect to which we differentiate, which will become important later when the same variable shows up in multiple function arguments.&lt;/p&gt;
-&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in &lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;&lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, essentially the custom primitive differentiation rule of &lt;code&gt;solve&lt;/code&gt;, is the elementary building block needed to derive sensitivities also in more complicated examples:&lt;/p&gt;
 &lt;p&gt;Consider a loss depending on the state $x(s_i)$ at fix time points $s_i$ through loss functions $L_i^{\text{exp}}$,&lt;/p&gt;
 &lt;p&gt;$$
 L^{\text{exp}} = \sum_i L_i^{\text{exp}}(s_i, x(s_i), p).
 $$&lt;/p&gt;
 &lt;p&gt;Without contortions we can obtain the sensitivity of $L^{\text{exp}}$ in $p$ (or in $x_0$) using the tool we have. For those a bit familiar with automatic differentiation, this is perhaps easiest to see if we write $L^{\text{exp}}$ as pseudo code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1 = solve(t0, x0, s1, p)
-    L = L1(s1, x1, p)
-    x2 = solve(s1, x1, s2, p)
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\&lt;br&gt;
+\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\
 \frac{\text{d}\lambda_{p}(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(x(t), \rightarrow p, t)}{\text{d} p} - \frac{\text{d} L_i^{\text{exp}}(  x(t),\rightarrow p)}{\text{d} p}^\dagger \delta(t-s_i).
 \end{aligned}
 $$&lt;/p&gt;
@@ -957,24 +851,23 @@ $$&lt;/p&gt;
 $$&lt;/p&gt;
 &lt;p&gt;Now we can even properly define the &lt;code&gt;rrule&lt;/code&gt; of &lt;code&gt;solve&lt;/code&gt; in the sense of &lt;code&gt;DiffRules.jl&lt;/code&gt;.&lt;/p&gt;
 &lt;h3 id=&#34;explicit-events-1&#34;&gt;Explicit events&lt;/h3&gt;
-&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1- = solve(t0, x0, s1, p)
-    L = L1-(s1, x1-, p) # saved before affect
-    x1 = a(x1-, p, s1) # affect
-    L += L1(s1, x1, p) # saved after affect
-    x2- = solve(s1, x1, s2, p)
-    L += L2-(s2, x2-, p)
-    x2 = a(x2-, p, s2) # affect
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
+&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref2:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved before affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved after affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\&lt;br&gt;
+\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\
 \lambda_p({s_j}-) &amp;amp;= \lambda_p({s_j}) -  \lambda({s_j})^\dagger \frac{\text{d} a(x({s_j}-), \rightarrow p, {s_j}-)}{\text{d} p}
 \end{aligned}
 $$&lt;/p&gt;
@@ -988,27 +881,26 @@ $$&lt;/p&gt;
 &lt;p&gt;to put emphasis on this, or equivalently, compute for a unspecified function $L(t, x, p)$ the result of $\frac{\text{d}L}{\text{d}p}$ with&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\&lt;br&gt;
+\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\
 &amp;amp;= \frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;which indicates that changing $p$ influences $L$ both through changes in $\tau_1$ as well as changes in&lt;/p&gt;
 &lt;p&gt;$$\xi_1 = x(\tau_1-).$$&lt;/p&gt;
 &lt;p&gt;This case where we have a loss function $L = L_1$ depending on $\tau_1$, $x(\tau_1)$, and $p$
-was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;p&gt;Therefore, the sensitivity of the event time with respect to parameters $\frac{\text{d}\tau}{\text{d}p}$ must be taken into account.
-Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the 
-&lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
+Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the &lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\&lt;br&gt;
+\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The total derivative&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; inside the bracket is:
 $$
 \begin{aligned}
-\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\&lt;br&gt;
+\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;Since&lt;/p&gt;
@@ -1031,8 +923,8 @@ for the second term of $\dfrac{\text{d}\tau_1(p)}{\text{d}p}$.&lt;/p&gt;
 &lt;p&gt;We can now write the gradient as:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\&lt;br&gt;
-+&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\&lt;br&gt;
+\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\
++&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\
 +&amp;amp; \frac{\text{d}L(\rightarrow \tau_1(p), \text{solve}(t_0, x_0, \rightarrow \tau_1(p),  p), p)}{\text{d}\tau_1} \frac{\text{d} \tau_1(p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
@@ -1101,7 +993,7 @@ $$&lt;/p&gt;
 &lt;li&gt;If both limits contribute to the loss function, the contributions ${\color{red}\lambda_1}$ and ${\color{blue}\lambda_{a,1}}$ are added.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h4 id=&#34;generalization-several-events&#34;&gt;Generalization: several events&lt;/h4&gt;
-&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
+&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
 &lt;p&gt;$$
 \text{solve}(t_0, x_0, t, p)  = \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p),
 $$&lt;/p&gt;
@@ -1109,16 +1001,15 @@ $$&lt;/p&gt;
 &lt;p&gt;(A good exercise to get familiar with these type of arguments is to verify&lt;/p&gt;
 &lt;p&gt;$$\frac{\text{d}}{\text{d}s} \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p) = 0.)$$&lt;/p&gt;
 &lt;p&gt;Essentially, we know how to address several events already by considering a loss function&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss2(t0, x0, p)
-    tau1, xi1 = solve2(t0, x0, g, p)
-    L = L1(tau1, xi1, p)
-    tau2, xi2 = solve2(tau1, xi1, g, p)
-    L += L2(tau2, xi2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
 &lt;p&gt;Differentiating by hand, we have the following modification of the method from the previous section, which can be derived choosing a particular function $L$ in the previous section which incorporates the &lt;code&gt;solve&lt;/code&gt; from $\tau_1$ to $t_{\text{end}}$. Note&lt;/p&gt;
 &lt;p&gt;$$
 \lambda_{\tau_1}^0 = \frac{\text{d}(\text{solve}(\rightarrow \tau_1,  a(\xi_1,p), t_{\text{end}}, p))}{\text{d} \tau_1} = - f(a(\xi_1,p), p, \tau_1).
@@ -1150,43 +1041,38 @@ $$&lt;/p&gt;
 &lt;li&gt;Backpropagate $\lambda_J = {\color{red}\lambda_J} + {\color{blue}\lambda_{a,J}} + {\color{green}\lambda_{c,J}}$ to the next event time $\tau_{J-1}$ and iterate over the remaining $J-1$ events.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
-&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
-&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (
-&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$. &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$.&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the &lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
 $$
 \begin{aligned}
-\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\&lt;br&gt;
-&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\&lt;br&gt;
+\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\
+&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\
 &amp;amp;+ \frac{\text{d}f(t, x_1(t),x_2(\rightarrow t),\dots,x_N(t))}{\text{d}t} + \dots +  \frac{\text{d}f(t, x_1(t),x_2(t),\dots,x_N(\rightarrow t))}{\text{d}t}.
 \end{aligned}
-$$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+$$&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1196,8 +1082,7 @@ $$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-
       <pubDate>Fri, 02 Jul 2021 11:08:22 +0200</pubDate>
       <guid>https://frankschae.github.io/post/shadowing/</guid>
       <description>&lt;p&gt;In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &amp;ldquo;butterfly effect&amp;rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.&lt;/p&gt;
-&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
+&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:&lt;/p&gt;
 &lt;p&gt;$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -1211,57 +1096,57 @@ Under the assumption of ergodicity, $\langle g \rangle_∞$ only depends on $p$.
 &lt;p&gt;One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{pmatrix}
-\text{d}x \\&lt;br&gt;
-\text{d}y \\&lt;br&gt;
-\text{d}z \\&lt;br&gt;
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\&lt;br&gt;
-x(\rho-z) - y\\&lt;br&gt;
-x y - \beta z \\&lt;br&gt;
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
-
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
-
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Statistics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;QuadGK&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SparseArrays&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# simulate 1 trajectory of the Lorenz system forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;28.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;initial&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;attractor&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &amp;quot;initial&amp;quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&amp;quot;attractor&amp;quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -1273,79 +1158,71 @@ savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.&lt;/p&gt;
-&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
+&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
 &lt;p&gt;$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function compute_objective(sol)
-  quadgk(t-&amp;gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&amp;quot;z(t)&amp;quot;, xlabel=L&amp;quot;t&amp;quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&amp;quot;\langle z\rangle&amp;quot;, lw = 2)
-savefig(pl2, &amp;quot;zsingle.png&amp;quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
-
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;quadgk&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;atol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;mean_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zsingle.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# for each value of the parameter, solve 20 times the initial value problem&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# wrap the procedure inside a function depending on p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ribbon&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;ho&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zvsrho.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;margin&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;600&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;300&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;z.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We obtain:&lt;/p&gt;
 
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&amp;quot;\langle z\rangle&amp;quot;, xlabel=L&amp;quot;\rho&amp;quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &amp;quot;zvsrho.png&amp;quot;)
 
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &amp;quot;z.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We obtain:&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -1357,40 +1234,30 @@ savefig(pl4, &amp;quot;z.png&amp;quot;)
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/z.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/z.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;z.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$&lt;/p&gt;
 &lt;h2 id=&#34;conventional-forward-mode-sensitivity-analysis-and-finite-differencing&#34;&gt;Conventional forward-mode sensitivity analysis and finite-differencing&lt;/h2&gt;
-&lt;p&gt;For non-chaotic systems, we would just use the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we find diverging values:&lt;/p&gt;
+&lt;p&gt;For non-chaotic systems, we would just use the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_forward&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_calculus&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we find diverging values:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\&lt;br&gt;
+&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$&lt;/p&gt;
@@ -1398,44 +1265,44 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$&lt;/p&gt;
-&lt;p&gt;Similarly, using 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-     xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&amp;quot;Float32&amp;quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
+&lt;p&gt;Similarly, using &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;500&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_Floats.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&amp;quot;Float32&amp;quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
 
-savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -1446,19 +1313,10 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Without animation:&lt;/p&gt;
 
@@ -1466,8 +1324,6 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-  
-
 
 
 
@@ -1478,60 +1334,51 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; &gt;
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_Floats.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;Luckily, the 
-&lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
+&lt;p&gt;Luckily, the &lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
 &lt;blockquote&gt;
 &lt;p&gt;Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&amp;ldquo;shadows&amp;rdquo;) the numerically computed one.&lt;/p&gt;
 &lt;/blockquote&gt;
 &lt;h2 id=&#34;shadowing-methods&#34;&gt;Shadowing methods&lt;/h2&gt;
-&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref2:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref3:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;lss-and-nilss-for-the-lorenz-system&#34;&gt;LSS and NILSS for the Lorenz system&lt;/h2&gt;
-&lt;p&gt;Switching to LSS or NILSS within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
+&lt;p&gt;Switching to LSS or NILSS within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# objective&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# LSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;CosWindowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0095888187322035&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Cos2Windowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0343951385924328&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0284286902740765&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_adjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.028428690274077&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# select via sensealg in solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9694728321500617&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;&lt;code&gt;CosWindowing()&lt;/code&gt; (default)&lt;/li&gt;
 &lt;li&gt;&lt;code&gt;Cos2Windowing()&lt;/code&gt;&lt;/li&gt;
@@ -1540,72 +1387,68 @@ dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
 &lt;p&gt;Additionally, an adjoint implementation &lt;code&gt;AdjointLSS()&lt;/code&gt; is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that &lt;code&gt;AdjointLSS(alpha=10.0)&lt;/code&gt; agrees perfectly with &lt;code&gt;ForwardLSS(alpha=10.0)&lt;/code&gt;. In all cases considered, we find the expected sensitivity value of $\approx 1$.&lt;/p&gt;
 &lt;p&gt;However, the use of &lt;code&gt;LSS()&lt;/code&gt; is (typically) much more expensive than the use of &lt;code&gt;NILSS()&lt;/code&gt;, because &lt;code&gt;LSS()&lt;/code&gt; needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of &lt;code&gt;NILSS()&lt;/code&gt; scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making &lt;code&gt;NILSS()&lt;/code&gt; more efficient.&lt;/p&gt;
 &lt;p&gt;In the &lt;code&gt;NILSS()&lt;/code&gt; algorithm, the user can control the number of steps per segment as well as the number of segments.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# NILSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# make sure trajectory is fully on the attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;120.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9966924374966089&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0416028730638789&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Warning: Detected a large value of ξ at the beginning of a segment.&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref4:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;NILSAS&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;li&gt;FD-NILSS&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref4:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019) &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019)&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019) &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019)&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1614,21 +1457,11 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
       <link>https://frankschae.github.io/post/hybridde/</link>
       <pubDate>Wed, 16 Jun 2021 14:50:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/hybridde/</guid>
-      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
+      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
 &lt;p&gt;This first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).&lt;/p&gt;
-&lt;p&gt;Note: Please check the 
-&lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
+&lt;p&gt;Note: Please check the &lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
 &lt;h2 id=&#34;hybrid-differential-equations&#34;&gt;Hybrid Differential Equations&lt;/h2&gt;
-&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
+&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
 &lt;p&gt;Some relevant examples for hybrid DEs with discrete or continuous callbacks are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; (&lt;code&gt;ContinuousCallback&lt;/code&gt;).&lt;/li&gt;
@@ -1638,7 +1471,7 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
 &lt;li&gt;pharmacokinetic models&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, where explicit dosing times change the drug concentration in the blood (&lt;code&gt;DiscreteCallback&lt;/code&gt;). The simplest possible example being the one-compartment model.&lt;/li&gt;
 &lt;li&gt;kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (&lt;code&gt;DiscreteCallback&lt;/code&gt;).&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;kicked-harmonic-oscillator&#34;&gt;Kicked harmonic oscillator&lt;/h2&gt;
 &lt;p&gt;Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form&lt;/p&gt;
 &lt;p&gt;$$
@@ -1647,40 +1480,37 @@ $$&lt;/p&gt;
 &lt;p&gt;where $a=0.1$ and $b=1$ with initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-x(t=0) &amp;amp;= 1  \\&lt;br&gt;
+x(t=0) &amp;amp;= 1  \\
 v(t=0) &amp;amp;= \dot{x}(t=0) = 0.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;This second-order ODE can be 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use 
-&lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity
-using Zygote
-using Random
-u0 = Float32[1.; 0.]
-
-tspan = (0.0f0,50.0f0)
-
-dtsave = 0.5f0
-t = tspan[1]:dtsave:tspan[2]
-
-function oscillator!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  return nothing
-end
-
-prob_data = ODEProblem(oscillator!,u0,tspan)
-
-# ODE without kicks
-pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
+&lt;p&gt;This second-order ODE can be &lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use &lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqFlux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Optim&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# ODE without kicks&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -1692,44 +1522,35 @@ pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &am
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as &lt;code&gt;1&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;kicktimes = tspan[1]:1:tspan[2]
-function kick!(integrator)
-  integrator.u[end] += one(eltype(integrator.u))
-end
-cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false))
-
-sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t)
-t_data = sol_data.t
-ode_data = Array(sol_data)
-
-# visualize data
-pl1 = plot(t_data,ode_data[1,:],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl1,t_data,ode_data[2,:],label=&amp;quot;data v(t)&amp;quot;)
-
-pl2 = plot(t_data[1:20],ode_data[1,1:20],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl2,t_data[1:20],ode_data[2,1:20],label=&amp;quot;data v(t)&amp;quot;)
-pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualize data&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
 
 
 
-  
 
 
 
@@ -1741,63 +1562,55 @@ pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting &lt;code&gt;save_positions=(true,true)&lt;/code&gt;, the kicks would be saved before &lt;strong&gt;and&lt;/strong&gt; after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;, we now aim at learning (potentially) missing parts of the model from these data traces.&lt;/p&gt;
 &lt;h3 id=&#34;high-domain-knowledge&#34;&gt;High domain knowledge&lt;/h3&gt;
 &lt;p&gt;For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the &lt;code&gt;affect!&lt;/code&gt; function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of &lt;code&gt;integrator.u&lt;/code&gt;. If we further set the initial value of that component to &lt;code&gt;1&lt;/code&gt;, then the neural network only has to learn that &lt;code&gt;du[3]&lt;/code&gt; is &lt;code&gt;0&lt;/code&gt;. In other words, the output of the neural network must be &lt;code&gt;0&lt;/code&gt; for all states &lt;code&gt;u&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;Random.seed!(123)
-nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1))
-p_nn1 = initial_params(nn1)
-
-function f1!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3] = nn1(u[1:2], p)[1]
-  return nothing
-end
-
-affect!(integrator) = integrator.u[2] += integrator.u[3]
-cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false))
-z0 = Float32[u0;one(u0[1])]
-prob1 = ODEProblem(f1!,z0,tspan,p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# to visualize the predictions of the trained neural network below
-function visualize(prob,p)
-  _prob = remake(prob,p=p)
-  ode_pred = Array(solve(_prob,Tsit5(),callback=cb,
-                 saveat=dtsave))[1:2,:]
-  pl1 = plot(t_data,ode_pred[1,:],label=&amp;quot;x(t)&amp;quot;)
-  scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=&amp;quot;data x(t)&amp;quot;)
-  pl2 = plot(t_data,ode_pred[2,:],label=&amp;quot;v(t)&amp;quot;)
-  scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=&amp;quot;data v(t)&amp;quot;)
-
-  pl = plot(pl1, pl2, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-  return pl, sum(abs2,ode_data .- ode_pred)
-end
-
-pl = plot(solve(prob1,Tsit5(),saveat=t,
-  callback=cb
-  ),label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;123&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FastChain&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tanh&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial_params&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# to visualize the predictions of the trained neural network below&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                 &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -1809,76 +1622,65 @@ pl = plot(solve(prob1,Tsit5(),saveat=t,
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;untrained_nn.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;which (of course) doesn&amp;rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### loss function
-function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint())
-  _prob = remake(prob,p=p)
-  pred = Array(solve(_prob,Tsit5(),callback=cb,
-               saveat=dtsave,sensealg=sensealg))[1:2,:]
-  sum(abs2,ode_data .- pred)
-end
-
-loss(p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The recently implemented tools are deeply hidden within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., 
-&lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;###################################
-# training loop
-# optimize the parameters for a few epochs with ADAM
-function train(prob, p_nn; sensealg=BacksolveAdjoint())
-  opt = ADAM(0.0003f0)
-  list_plots = []
-  losses = []
-  for epoch in 1:200
-    println(&amp;quot;epoch: $epoch / 200&amp;quot;)
-    _dy, back = Zygote.pullback(p -&amp;gt; loss(p,
-      prob=prob,
-      sensealg=sensealg), p_nn)
-    gs = @time back(one(_dy))[1]
-    push!(losses, _dy)
-    if epoch % 10 == 0
-      # plot every xth epoch
-      pl, test_loss = visualize(prob, p_nn)
-      println(&amp;quot;Loss (epoch: $epoch): $test_loss&amp;quot;)
-      display(pl)
-      push!(list_plots, pl)
-    end
-    Flux.Optimise.update!(opt, p_nn, gs)
-    println(&amp;quot;&amp;quot;)
-  end
-  return losses, list_plots
-end
-
-# plot training loss
-losses, list_plots = train(prob1, p_nn1)
-pl1 = plot(losses, lw = 1.5, xlabel = &amp;quot;epoch&amp;quot;, ylabel=&amp;quot;loss&amp;quot;, legend=false)
-pl2 = list_plots[end]
-pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t,
-   callback=cb
-  ), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-
-pl = plot(pl2,pl3)
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### loss function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ReverseDiffAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;               &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The recently implemented tools are deeply hidden within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., &lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;###################################&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# training loop&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# optimize the parameters for a few epochs with ADAM&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ADAM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0003f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;200&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt; / 200&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;%&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;c&#34;&gt;# plot every xth epoch&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_loss&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Loss (epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;): &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$test_loss&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;display&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Optimise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;update!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot training loss&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;epoch&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;loss&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -1890,98 +1692,85 @@ pl = plot(pl2,pl3)
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/trained1.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/trained1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;trained1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We see the expected constant value of &lt;code&gt;u[3]&lt;/code&gt;, indicating a kick to the velocity of &lt;code&gt;+=1&lt;/code&gt;, at the kicking times over the full time interval.&lt;/p&gt;
 &lt;h2 id=&#34;reducing-the-domain-knowledge&#34;&gt;Reducing the domain knowledge&lt;/h2&gt;
-&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
+&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;changing the initial condition of the third component of &lt;code&gt;u&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;using another affect function &lt;code&gt;affect!(integrator) = integrator.u[2] = integrator.u[3]&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;dropping the knowledge that only &lt;code&gt;u[2]&lt;/code&gt; gets a kick by using a neural network with &lt;code&gt;2&lt;/code&gt; outputs (+ a fourth component in the ODE):&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4]
-function f2!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;fitting the parameters $a$ and $b$ simultaneously:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f3!(du,u,p,t)
-  a = p[end-1]
-  b = p[end]
-  nn_weights = p[1:end-2]
-
-  du[1] = u[2]
-  du[2] = -b*u[1] - a*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f3!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;inferring the entire underlying dynamics using a neural network with &lt;code&gt;4&lt;/code&gt; outputs:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f4!(du,u,p,t)
-  Ω = nn3(u[1:2], p)
-
-  du[1] = Ω[1]
-  du[2] = Ω[2]
-  du[3:4] .= Ω[3:4]
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f4!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;etc.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;refine the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;) and&lt;/li&gt;
-&lt;li&gt;support direct usage through the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
+&lt;li&gt;support direct usage through the &lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021) &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021)&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
diff --git a/tag/gsoc-2021/page/1/index.html b/tag/gsoc-2021/page/1/index.html
index 39ca607..c25977b 100644
--- a/tag/gsoc-2021/page/1/index.html
+++ b/tag/gsoc-2021/page/1/index.html
@@ -1 +1,10 @@
-<!DOCTYPE html><html><head><title>https://frankschae.github.io/tag/gsoc-2021/</title><link rel="canonical" href="https://frankschae.github.io/tag/gsoc-2021/"/><meta name="robots" content="noindex"><meta charset="utf-8" /><meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/gsoc-2021/" /></head></html>
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diff --git a/tag/high-weak-order-solver/index.html b/tag/high-weak-order-solver/index.html
index 4294abd..a5497bd 100644
--- a/tag/high-weak-order-solver/index.html
+++ b/tag/high-weak-order-solver/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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-
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-  <h1>High weak order solver</h1>
+  <h1>High Weak Order Solver</h1>
 
   
 
@@ -552,102 +758,321 @@ <h1>High weak order solver</h1>
 <div class="universal-wrapper">
   
 
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+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/high-weak/" >High weak order SDE solvers</a>
+    </div>
+
     
-    
-    <div>
-      <h2><a href="/post/high-weak/" >High weak order SDE solvers</a></h2>
+    <a href="/post/high-weak/"  class="summary-link">
       <div class="article-style">
-        
-          This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
+        This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
 After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
-        
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+    </a>
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+  <span class="article-date">
     
     
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+          Last updated on
+      
     
-    <div>
-      <h2><a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a></h2>
-      <div class="article-style">
-        
-          First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
-        
-      </div>
-    </div>
+    Aug 18, 2020
+  </span>
   
 
   
 
+  
+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    10 min read
+  </span>
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+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a>
+    </div>
+
+    
+    <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/"  class="summary-link">
+      <div class="article-style">
+        First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
 
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diff --git a/tag/high-weak-order-solver/index.xml b/tag/high-weak-order-solver/index.xml
index 5e2e671..2941f52 100644
--- a/tag/high-weak-order-solver/index.xml
+++ b/tag/high-weak-order-solver/index.xml
@@ -1,14 +1,14 @@
 <?xml version="1.0" encoding="utf-8" standalone="yes" ?>
 <rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
   <channel>
-    <title>High weak order solver | FS</title>
+    <title>High Weak Order Solver | FS</title>
     <link>https://frankschae.github.io/tag/high-weak-order-solver/</link>
       <atom:link href="https://frankschae.github.io/tag/high-weak-order-solver/index.xml" rel="self" type="application/rss+xml" />
-    <description>High weak order solver</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Mon, 17 Aug 2020 14:46:35 +0200</lastBuildDate>
+    <description>High Weak Order Solver</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Mon, 17 Aug 2020 14:46:35 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
-      <title>High weak order solver</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>High Weak Order Solver</title>
       <link>https://frankschae.github.io/tag/high-weak-order-solver/</link>
     </image>
     
@@ -17,11 +17,8 @@
       <link>https://frankschae.github.io/post/high-weak/</link>
       <pubDate>Mon, 17 Aug 2020 14:46:35 +0200</pubDate>
       <guid>https://frankschae.github.io/post/high-weak/</guid>
-      <description>&lt;p&gt;This post summarizes our new 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
-for the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
+      <description>&lt;p&gt;This post summarizes our new &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
+for the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
 &lt;p&gt;After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
 Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics.
 Finally, we discuss how to run simulations on GPU hardware.&lt;/p&gt;
@@ -52,16 +49,14 @@ $$
 \rm{E}(|X(T) -Y(T)|) \leq C \cdot h^p,
 $$
 for each $h \in [0, \delta]$.&lt;/p&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
+&lt;p&gt;The &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
 for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms.
 For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI)
 can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where &lt;em&gt;b&lt;/em&gt; has only entries on its diagonal or $m=1$.
 The main difficulty for the construction of strong methods with an order &amp;gt; 1/2 arises from the need of an accurate estimation of
 multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of &lt;em&gt;dW&lt;/em&gt; in the case
 of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed
-in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also 
-&lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
+in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also &lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;weak-convergence&#34;&gt;Weak convergence&lt;/h2&gt;
 &lt;p&gt;Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt; in many situations. In these cases, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient and &amp;ndash; due to the less restrictive formulation of the objective &amp;ndash; those solvers are computationally cheaper than their strong counterparts.
 For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics.
@@ -76,19 +71,17 @@ $~$&lt;/p&gt;
 &lt;p&gt;for any polynomial $g$ for each $h \in [0, \delta]$.&lt;/p&gt;
 &lt;p&gt;We demonstrate below that &lt;strong&gt;high weak order solvers&lt;/strong&gt; are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.&lt;/p&gt;
 &lt;h2 id=&#34;new-high-weak-order-methods&#34;&gt;New high weak order methods&lt;/h2&gt;
-&lt;p&gt;A list of all new weak solvers is available in the 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
+&lt;p&gt;A list of all new weak solvers is available in the &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
 Note that we also implemented methods designed for the Stratonovich sense.
 For the subsequent examples regarding Ito SDEs, we use only a subset of the plethora of second-order weak solvers.
-We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
-We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
+We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
+We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref2:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref3:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
 The latter is the simplest weak solver, where the Gaussian increments of the strong Euler-Maruyama scheme are replaced by
 two-point distributed random variables with similar moment properties.&lt;/p&gt;
 &lt;p&gt;Rößler&amp;rsquo;s SRK schemes are particularly designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;, since only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn and since the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.
 &lt;code&gt;PL1WM()&lt;/code&gt; in contrast needs to simulate &lt;code&gt;m(m+1)/2&lt;/code&gt; random variables but a smaller number of order conditions needs to be fulfilled.&lt;/p&gt;
 &lt;h3 id=&#34;convergence-tests&#34;&gt;Convergence tests&lt;/h3&gt;
-&lt;p&gt;As in the 
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As in the &lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -99,63 +92,61 @@ $$ ~$$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;for the function $f(x)=(x_1)^2$, which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e6&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-using Test
-using Random
-using Plots
-using DiffEqDevTools
-
-function prob_func(prob, i, repeat)
-    remake(prob,seed=seeds[i])
-end
-
-u₀ = [1.0,1.0]
-function f1!(du,u,p,t)
-  @inbounds begin
-    du[1] = -273//512*u[1]
-    du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-  end
-  return nothing
-end
-function g1!(du,u,p,t)
-  @inbounds begin
-    du[1,1] = 1//4*u[1]
-    du[1,2] = 1//16*u[1]
-    du[2,1] = (1-2*sqrt(2))/4*u[1]
-    du[2,2] = 1//10*u[1]+1//16*u[2]
-  end
-  return nothing
-end
-dts = 1 .//2 .^(3: -1:0)
-tspan = (0.0,3.0)
-
-h2(z) = z^2 # but apply it only to u[1]
-
-prob = SDEProblem(f1!,g1!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-numtraj = Int(1e6)
-seed = 100
-Random.seed!(seed)
-seeds = rand(UInt, numtraj)
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h2(sol[end][1]),false),
-        prob_func = prob_func
-        )
-
-sim = test_convergence(dts,ensemble_prob,DRI1(),
-  save_everystep=false,trajectories=numtraj,save_start=false,adaptive=false,
-  weak_timeseries_errors=false,weak_dense_errors=false,
-  expected_value=exp(-3.0)
-  )
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqDevTools&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# but apply it only to u[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Int&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_convergence&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
 Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function for the other aforementioned solvers, we obtain the convergence plot:&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -167,18 +158,12 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/weak_conv.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/weak_conv.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;weak_conv.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Note that the &lt;code&gt;SimplifiedEM&lt;/code&gt; and the &lt;code&gt;EM&lt;/code&gt; scheme fall on top of each other.
 &lt;code&gt;DRI1()&lt;/code&gt; achieves the smallest errors for a fixed &lt;code&gt;dt&lt;/code&gt; in this study.&lt;/p&gt;
@@ -186,35 +171,29 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 &lt;p&gt;Ultimately, we are not only interested in the general convergence slope of an algorithm but also in its speed.
 We&amp;rsquo;d like to select the fastest method depending on the permitted tolerance.
 This is commonly studied using work-precision diagram.
-Thanks to 
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;reltols = 1.0 ./ 4.0 .^ (1:4)
-abstols = reltols#[0.0 for i in eachindex(reltols)]
-setups = [
-          Dict(:alg=&amp;gt;DRI1(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;PL1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;EM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;SimplifiedEM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI2WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false)
-          ]
-test_dt = 1//10000
-appxsol_setup = Dict(:alg=&amp;gt;EM(), :dt=&amp;gt;test_dt)
-wp = @time WorkPrecisionSet(ensemble_prob,
-                            abstols,reltols,setups,test_dt;
-                            maxiters = 1e7,verbose=false,
-                            save_everystep=false,save_start=false,
-                            appxsol_setup = appxsol_setup,
-                            expected_value=exp(-3.0),
-                            trajectories=numtraj, error_estimate=:weak_final)
-plt = plot(wp;legend=:bottomleft)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
+Thanks to &lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;./&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;4.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;c&#34;&gt;#[0.0 for i in eachindex(reltols)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;PL1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;SimplifiedEM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI2WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10000&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WorkPrecisionSet&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;maxiters&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;verbose&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;error_estimate&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:weak_final&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-  
 
 
 
@@ -226,88 +205,87 @@ plt = plot(wp;legend=:bottomleft)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;WorkPrecision.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Therefore, &lt;code&gt;DRI1&lt;/code&gt; has the best performance in this non-commutative noise case if the error is supposed to stay below 1e-3.
 For larger permitted errors, the &lt;code&gt;SimplifiedEM&lt;/code&gt; scheme might be a good choice. However, the first order methods
 are outclassed when high precision is more of a concern.
-We plan to perform more in-depth benchmarks in the near future what will be reported on the 
-&lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
+We plan to perform more in-depth benchmarks in the near future what will be reported on the &lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
 &lt;h3 id=&#34;adaptive-step-size-control&#34;&gt;Adaptive step-size control&lt;/h3&gt;
-&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
+&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
 &lt;p&gt;To use adaptive step-size control, it is sufficient to set &lt;code&gt;adaptive=true&lt;/code&gt; (default setting). Optionally, one may also pass absolute and relative tolerances.&lt;/p&gt;
 &lt;p&gt;The following julia code simulates the stochastic version of the Brusselator equations with intitial condition&lt;/p&gt;
 &lt;p&gt;$$ X(t=0)=  \begin{pmatrix} 0.1 \\ 0\end{pmatrix},$$&lt;/p&gt;
 &lt;p&gt;on a time span $\mathbb{I}=[0, 100]$ for adaptive (&lt;code&gt;sol&lt;/code&gt;) and fixed step sizes (&lt;code&gt;sol_na&lt;/code&gt;):&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq, DiffEqNoiseProcess, Random
-using Plots
-using DiffEqGPU
-
-function prob_func(prob, i, repeat)
-  Random.seed!(seeds[i])
-  W = WienerProcess(0.0,0.0,0.0)
-  remake(prob,noise=W)
-end
-
-function brusselator_f!(du,u,p,t)
-  @inbounds begin
-    du[1] = (p[1]-1)*u[1]+p[1]*u[1]^2+(u[1]+1)^2*u[2]
-    du[2] = -p[1]*u[1]-p[1]*u[1]^2-(u[1]+1)^2*u[2]
-  end
-  nothing
-end
-
-function scalar_noise!(du,u,p,t)
-  @inbounds begin
-    du[1] = p[2]*u[1]*(1+u[1])
-    du[2] = -p[2]*u[1]*(1+u[1])
-   end
-   nothing
-end
-
-
-# fix seeds
-seed = 100
-Random.seed!(seed)
-numtraj= 100
-seeds = rand(UInt, numtraj)
-W = WienerProcess(0.0,0.0,0.0)
-
-
-u0 = [-0.1f0,0.0f0]
-tspan = (0.0f0,100.0f0)
-p = [1.9f0,0.1f0]
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqGPU&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# fix seeds&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.9f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleSummary&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fillalpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;time t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;X(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
-prob = SDEProblem(brusselator_f!,scalar_noise!,u0,tspan,p,noise=W)
 
-ensembleprob = EnsembleProblem(prob, prob_func = prob_func)
 
-sol = @time solve(ensembleprob,DRI1(),dt=0.1,EnsembleCPUArray(),trajectories=numtraj)
-sol_na = @time solve(ensembleprob,DRI1(),dt=0.8,adaptive=false,EnsembleCPUArray(),trajectories=numtraj)
 
 
-summ = EnsembleSummary(sol,0.0f0:0.5f0:100f0)
-pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quot;X(t)&amp;quot;, label= [&amp;quot;x₁(t)&amp;quot; &amp;quot;x₂(t)&amp;quot;], legend=true)
 
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -315,72 +293,55 @@ pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quo
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator_many_trajectories.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
+&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref3:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
 as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DifferentialEquations.EnsembleAnalysis
-meansol = timeseries_steps_mean(sol)
-meansol_na = timeseries_point_mean(sol_na,meansol.t)
-dts = []
-tmp1 = tspan[1]
-for tmp2 in meansol.t
-  global tmp1
-  push!(dts,tmp2-tmp1)
-  tmp1 = tmp2
-end
-#
-
-list_plots = []
-for i in 1:length(meansol.u)
-  l = @layout [a  b]
-  plt1 = plot(meansol[1, 1:i],meansol[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;adaptive&amp;quot;,
-         lw=2,
-         linecolor=1
-         )
-  plot!(meansol_na[1, 1:i],meansol_na[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;fixed step size&amp;quot;,
-         lw=2,
-         linecolor=2
-         )
-  pl2 = scatter(dts[1:i], xlabel = &amp;quot;step&amp;quot;, yaxis= &amp;quot;dtᵢ&amp;quot;, xlim = (0, length(meansol.u)),  ylim = (0.0, 2.3), legend=false)
-  plt = plot(plt1, pl2, layout = l)
-  push!(list_plots, plt)
-end
-
-anim = animate(list_plots,lw=2,every=1)
-
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleAnalysis&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_steps_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_point_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;global&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;#&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@layout&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;adaptive&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;fixed step size&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scatter&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;step&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;dtᵢ&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)),&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;2.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -392,53 +353,46 @@ anim = animate(list_plots,lw=2,every=1)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gpu-usage&#34;&gt;GPU usage&lt;/h3&gt;
 &lt;p&gt;All necessary tools to accelerate the simulation of (stochastic) differential equations on GPUs within the SciML
-ecosystem are collected in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
+ecosystem are collected in the &lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;Currently, bounds checking and return values are not allowed, i.e., functions must be
 written in the form:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f!(du,u,p,t)
-  @inbounds begin
-    du[1] = ..
-  end
-  nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;..&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
 Note that for some high weak order solvers GPU usage is not recommended as scalar indexing is used.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -447,21 +401,14 @@ Note that for some high weak order solvers GPU usage is not recommended as scala
       <link>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</link>
       <pubDate>Sat, 30 May 2020 15:10:33 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</guid>
-      <description>&lt;p&gt;First and foremost, I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
+      <description>&lt;p&gt;First and foremost, I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
 Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both  &lt;strong&gt;new high weak order solvers&lt;/strong&gt; as well as &lt;strong&gt;adjoint sensitivity methods&lt;/strong&gt; for stochastic differential equations (SDEs).
-The project is embedded within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
-Ultimately, the planned contributions will allow researchers to simulate (or even 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
+The project is embedded within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
+Ultimately, the planned contributions will allow researchers to simulate (or even &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
 Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.&lt;/p&gt;
 &lt;p&gt;This post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.&lt;/p&gt;
 &lt;h2 id=&#34;high-weak-order-solvers&#34;&gt;High Weak Order Solvers&lt;/h2&gt;
-&lt;p&gt;Currently, the 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
+&lt;p&gt;Currently, the &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
 In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.&lt;/p&gt;
 &lt;p&gt;However in many situations, we are only aiming for computing an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt;.
 In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods.
@@ -469,7 +416,7 @@ In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&
 As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.&lt;/p&gt;
 &lt;h3 id=&#34;second-order-runge-kutta-methods-for-ito-sdes&#34;&gt;Second order Runge-Kutta methods for Ito SDEs&lt;/h3&gt;
 &lt;p&gt;In the beginning of the community bonding period I finished the implementations of the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and &lt;code&gt;RI1()&lt;/code&gt;&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;. Specifically, only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn (in contrast to &lt;code&gt;m(m+1)/2&lt;/code&gt; from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.&lt;/p&gt;
-&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -478,47 +425,40 @@ $$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e7&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;. The code for a single  &lt;code&gt;dt&lt;/code&gt; reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-numtraj = 1e7
-u₀ = [1.0,1.0]
-function f!(du,u,p,t)
-  du[1] = -273//512*u[1]
-  du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-end
-function g!(du,u,p,t)
-  du[1,1] = 1//4*u[1]
-  du[1,2] = 1//16*u[1]
-  du[2,1] = (1-2*sqrt(2))/4*u[1]
-  du[2,2] = 1//10*u[1]+1//16*u[2]
-end
-dt = 1//8
-tspan = (0.0,10.0)
-prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-h(z) = z^2
-
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h(sol[end][1]),false)
-        )
-sol = solve(ensemble_prob, DRI1();
-        dt=dt,
-        save_start=false,
-        save_everystep=false,
-        weak_timeseries_errors=false,
-        weak_dense_errors=false,
-        trajectories=numtraj)
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
-&lt;pre&gt;&lt;code&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;();&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
 
 
-  
 
 
 
@@ -530,29 +470,24 @@ sol = solve(ensemble_prob, DRI1();
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/DRI1.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/DRI1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;DRI1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In the next couple of weeks, my focus will be on&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;adding other high weak order solvers,&lt;/li&gt;
 &lt;li&gt;implementing adaptive time stepping.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;More of our near-term goals are collected in this 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;More of our near-term goals are collected in this &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;adjoint-sensitivity-methods-for-sdes&#34;&gt;Adjoint Sensitivity Methods for SDEs&lt;/h2&gt;
-&lt;p&gt;In 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
+&lt;p&gt;In &lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
 &lt;p&gt;$$
 L = |f(p)-y|^2.
 $$&lt;/p&gt;
@@ -561,46 +496,43 @@ One possibility to obtain the gradient information for (stochastic) differential
 While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.&lt;/p&gt;
 &lt;p&gt;Alternatively to the &amp;ldquo;direct&amp;rdquo; AD approaches, the &lt;strong&gt;adjoint sensitivity method&lt;/strong&gt; can be used&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.
 Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).&lt;/p&gt;
-&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
+&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
 &lt;h3 id=&#34;reverse-sde-time-evolution&#34;&gt;Reverse SDE time evolution&lt;/h3&gt;
 &lt;p&gt;The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms.
 As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed.
 In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values.
-After some fixes for the 
-&lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
+After some fixes for the &lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
 &lt;p&gt;As an example, the code below computes first the forward evolution of an SDE&lt;/p&gt;
 &lt;p&gt;$$ dX  =  \alpha X dt + \beta X dW$$&lt;/p&gt;
 &lt;p&gt;with $\alpha=1.01$, $\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;  using StochasticDiffEq, DiffEqNoiseProcess
-
-  α=1.01
-  β=0.87
-
-  dt = 1e-3
-  tspan = (0.0,1.0)
-  u₀=1/2
-
-  tarray =  collect(tspan[1]:dt:tspan[2])
-
-  f!(du,u,p,t) = du .= α*u
-  g!(du,u,p,t) = du .= β*u
-
-
-  prob = SDEProblem(f!,g!,[u₀],tspan)
-  sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false)
-
-  _sol = deepcopy(sol) # to make sure the plot is correct
-  W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W))
-  prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1)
-  sol1 = solve(prob1,EulerHeun(),dt=dt)
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# to make sure the plot is correct&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NoiseGrid&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -612,18 +544,11 @@ After some fixes for the
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/animation.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/animation.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;animation.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gradients-of-diagonal-sdes&#34;&gt;Gradients of diagonal SDEs&lt;/h3&gt;
 &lt;p&gt;I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.&lt;/p&gt;
@@ -637,82 +562,80 @@ $$&lt;/p&gt;
 &lt;p&gt;acting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\&lt;br&gt;
+\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\
 \frac{d L}{d \beta}  &amp;amp;= 2 \sum_i W(t_i) |X(t_i)|^2
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Test, LinearAlgebra
-using DiffEqSensitivity, StochasticDiffEq
-using Random
-
-seed = 100
-Random.seed!(seed)
-
-u₀ = [0.5]
-tstart = 0.0
-tend = 0.1
-dt = 0.005
-trange = (tstart, tend)
-t = tstart:dt:tend
-tarray = collect(t)
-
-function g(u,p,t)
-  sum(u.^2.0)
-end
-
-function dg!(out,u,p,t,i)
-  (out.=-2.0*u)
-end
-
-p2 = [1.01,0.87]
-
-f(u,p,t) = p[1]*u
-σ(u,p,t) = p[2]*u
-
-
-Random.seed!(seed)
-prob = SDEProblem(f,σ,u₀,trange,p2)
-sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true)
-res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint())
-
-
-noise = vec((@. sol.W(tarray)))
-Wextracted = [W[1][1] for W in noise]
-resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp = [resp1, resp2]
-
-@test isapprox(res_p&#39;, resp, rtol = 1e-6)
-# True
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.005&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:tend&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.=-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RKMil&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;interpretation&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:Stratonovich&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res_u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adjoint_sensitivities&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vec&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;isapprox&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# True&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;finish the current backsolve adjoint version,&lt;/li&gt;
 &lt;li&gt;allow for computing the gradients of non-commuting SDEs,&lt;/li&gt;
 &lt;li&gt;implement also an interpolation adjoint version,&lt;/li&gt;
 &lt;li&gt;benchmark it with respect to AD approaches&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
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-  <h1>Hybrid differential equations</h1>
+  <h1>Hybrid Differential Equations</h1>
 
   
 
@@ -552,101 +758,328 @@ <h1>Hybrid differential equations</h1>
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+      <a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a>
+    </div>
+
     
-    
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-      <h2><a href="/post/bouncing_ball/" >Sensitivity Analysis of Hybrid Differential Equations</a></h2>
+    <a href="/post/bouncing_ball/"  class="summary-link">
       <div class="article-style">
-        
-          In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
-        
+        In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces.
       </div>
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+      Frank Schäfer and Moritz Schauer</span>
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-      <h2><a href="/post/hybridde/" >Neural Hybrid Differential Equations</a></h2>
-      <div class="article-style">
-        
-          I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
-        
-      </div>
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+      
+    
+    Jul 16, 2021
+  </span>
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+      <a href="/post/hybridde/" >Neural Hybrid Differential Equations</a>
+    </div>
+
+    
+    <a href="/post/hybridde/"  class="summary-link">
+      <div class="article-style">
+        I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
+      </div>
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-</footer>
 
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@@ -735,7 +1166,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
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-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
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         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -750,5 +1182,25 @@ <h5 class="modal-title">Cite</h5>
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diff --git a/tag/hybrid-differential-equations/index.xml b/tag/hybrid-differential-equations/index.xml
index c41f6f2..a9205e3 100644
--- a/tag/hybrid-differential-equations/index.xml
+++ b/tag/hybrid-differential-equations/index.xml
@@ -1,14 +1,14 @@
 <?xml version="1.0" encoding="utf-8" standalone="yes" ?>
 <rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
   <channel>
-    <title>Hybrid differential equations | FS</title>
+    <title>Hybrid Differential Equations | FS</title>
     <link>https://frankschae.github.io/tag/hybrid-differential-equations/</link>
       <atom:link href="https://frankschae.github.io/tag/hybrid-differential-equations/index.xml" rel="self" type="application/rss+xml" />
-    <description>Hybrid differential equations</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Fri, 16 Jul 2021 13:24:04 +0200</lastBuildDate>
+    <description>Hybrid Differential Equations</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Fri, 16 Jul 2021 13:24:04 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
-      <title>Hybrid differential equations</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Hybrid Differential Equations</title>
       <link>https://frankschae.github.io/tag/hybrid-differential-equations/</link>
     </image>
     
@@ -20,96 +20,106 @@
       <description>&lt;p&gt;In this post, we discuss sensitivity analysis of differential equations with state changes caused by events triggered at defined moments, for example reflections, bounces off a wall or other sudden forces. These are described by hybrid differential equations&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. We highlight differences between explicit&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and implicit events&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;. As a paradigmatic example, we consider a bouncing ball described by the ODE&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\&lt;br&gt;
+\text{d}z(t) &amp;amp;= v(t) \text{d}t, \\
 \text{d}v(t) &amp;amp;= -\mathrm g\thinspace  \text{d}t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;with initial condition&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t=0) &amp;amp;= z_0 = 5, \\&lt;br&gt;
+z(t=0) &amp;amp;= z_0 = 5, \\
 v(t=0) &amp;amp;= v_0 = -0.1.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
 &lt;p&gt;The initial condition contains the initial height $z_0$ and initial velocity $v_0$ of the ball. We have two important parameters in this system. First, there is the gravitational constant $\mathrm g=10$ modeling the acceleration of the ball due to an approximately constant gravitational field.&lt;/p&gt;
-&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
+&lt;p&gt;Second, we model the ground as barrier at $z = 0$ where the ball bounces off in opposite direction. We include a dissipation factor $\gamma=0.8$ (&lt;a href=&#34;https://en.wikipedia.org/wiki/Coefficient_of_restitution&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;coefficient of restitution&lt;/a&gt;) that accounts for a imperfect elastic bounce on the ground.&lt;/p&gt;
 &lt;p&gt;When ignoring the bounces, we can straightforwardly integrate the ODE analytically&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\&lt;br&gt;
+z(t) &amp;amp;= z_0 + v_0 t - \frac{\mathrm g}{2} t^2, \\
 v(t) &amp;amp;= v_0 - \mathrm g\thinspace  t
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;or numerically using the OrdinaryDiffEq package from the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### simulate forward
+&lt;p&gt;or numerically using the OrdinaryDiffEq package from the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### simulate forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# dynamics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# parameters and solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;5.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.9&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;γ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward_no_bounce.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
-using ForwardDiff, Zygote, OrdinaryDiffEq, DiffEqSensitivity
-using Plots, LaTeXStrings
 
-# dynamics
-function f(du,u,p,t)
-  du[1] = u[2]
-  du[2] = -p[1]
-end
 
-# parameters and solve
-z0 = 5.0
-v0 = -0.1
-t0 = 0.0
-tend = 1.9
-g = 10
-γ = 0.8
 
-u0 = [z0,v0]
-tspan = (t0,tend)
-p = [g, γ]
-prob = ODEProblem(f,u0,tspan,p)
 
-# plot forward trajectory
-sol = solve(prob,Tsit5(),saveat=0.1)
-pl = plot(sol, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomleft)
-hline!(pl, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-savefig(pl,&amp;quot;BB_forward_no_bounce.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34;&gt;&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;https://i.imgur.com/fKTiDEe.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
+&lt;/p&gt;
 
 
 
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward_no_bounce.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+
+
+
+
+
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward_no_bounce.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Of course, this way the ball continues to fall through the barrier at $z=0$.&lt;/p&gt;
 &lt;h2 id=&#34;forward-simulation-with-events&#34;&gt;Forward simulation with events&lt;/h2&gt;
 &lt;p&gt;At time $\tau$ around $\tau \approx 1$, the ball hits the ground $z(\tau) = 0$, and is inelastically reflected while dissipating a fraction of its energy. This can be modeled by re-initializing the ODE at time $\tau$ with new initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau}) &amp;amp;=  z(\tau-) ,\\&lt;br&gt;
+z({\tau}) &amp;amp;=  z(\tau-) ,\\
 v({\tau})&amp;amp;= -\gamma v(\tau-) ,
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -126,15 +136,15 @@ $$&lt;/p&gt;
 &lt;p&gt;The full explicit trajectory $z_{\rm exp}(t) = z(t)$ is determined by&lt;/p&gt;
 &lt;p&gt;$$
 z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\&lt;br&gt;
--0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau, \\
+-0.4901 \mathrm g - 0.5 \mathrm g (-0.99005 + t)^2 + 0.99005 v_0 + z_0\\
 \quad  - (-0.99005 + t) (-0.99005 \mathrm g + v_0)\gamma  ,&amp;amp; \forall t \ge \tau,
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;where we used&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}&lt;br&gt;
-z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\&lt;br&gt;
+z({\tau})&amp;amp;=  z_0 + 0.99005 v_0 -0.4901 \mathrm g, \\
 v({\tau})&amp;amp;= -\gamma v({\tau-}) = -\gamma(v_0 - 0.99005 \mathrm g)  .
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
@@ -143,15 +153,14 @@ $$&lt;/p&gt;
 a(z,v) = (z, -\gamma v).
 $$&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;DiscreteCallback&lt;/code&gt; in this case to simulate the system.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with DiscreteCallback (explicit event)
-tstar = (v0 + sqrt(v0^2+2*z0*g))/g
-condition1(u,t,integrator) = (t == tstar)
-affect!(integrator) = integrator.u[2] = -integrator.p[2]*integrator.u[2]
-cb1 = DiscreteCallback(condition1,affect!,save_positions=(true,true))
-
-sol1 = solve(prob,Tsit5(),callback=cb1, saveat=0.1, tstops=[tstar])
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with DiscreteCallback (explicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;v0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiscreteCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstops&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Evidently, by choosing an explicit definition of the event, the impact time is fixed. The reflection event is triggered at $\tau = 0.99005$, a time where under different initial configurations the ball perhaps hasn’t reached the ground.&lt;/p&gt;
 &lt;h3 id=&#34;implicit-events&#34;&gt;Implicit events&lt;/h3&gt;
 &lt;p&gt;The physically more meaningful description of a bouncing ball is therefore given by an implicit description of the event in form of a condition (event function)&lt;/p&gt;
 &lt;p&gt;$$
@@ -160,7 +169,7 @@ $$&lt;/p&gt;
 &lt;p&gt;where an event occurs at time $\tau$ if $g(z(\tau),v(\tau),p,\tau) = 0$. We have already used this condition to define our impact time $\tau$ when modeling the bounce explicitly. The implicit formulation also lends itself to take multiple bounces into account by triggering the event every time  $g(z,v,p,t) = 0$.&lt;/p&gt;
 &lt;p&gt;As in the previous case, we can analytically compute the full trajectory of the ball. By substituting the formula for $\tau$ we have at the event time&lt;/p&gt;
 &lt;p&gt;\begin{aligned}
-z({\tau})&amp;amp;= 0, \\&lt;br&gt;
+z({\tau})&amp;amp;= 0, \\
 v({\tau}-)&amp;amp;= - \sqrt{v_0^2 + 2 \mathrm g z_0}
 \end{aligned}&lt;/p&gt;
 &lt;p&gt;for the left limit and&lt;/p&gt;
@@ -170,33 +179,28 @@ v({\tau})&amp;amp;= \gamma \sqrt{v_0^2 + 2 \mathrm g z_0}
 &lt;p&gt;right after the bounce. Thus, the full trajectory $z_{\rm imp}(t) = z(t)$ is given by&lt;/p&gt;
 &lt;p&gt;$$
 (\star) \quad z(t) = \begin{cases}
-z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\&lt;br&gt;
--\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\&lt;br&gt;
+z_0 + v_0 t - \dfrac{\mathrm g}{2} t^2 ,&amp;amp; \forall t &amp;lt; \tau ,\\
+-\dfrac{-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0}}{2 \mathrm g}  \\
 \quad\cdot \space (-\mathrm g t + v_0 + \sqrt{v_0^2 + 2 \mathrm g z_0} (1 + 2 \gamma)), &amp;amp; \forall t \ge \tau.
 \end{cases}
 $$&lt;/p&gt;
 &lt;p&gt;This is correct even if one substitutes, e.g., a value with higher precision $\mathrm g = 9.81$ for the gravitation constant.&lt;/p&gt;
 &lt;p&gt;Numerically, we use a &lt;code&gt;ContinuousCallback&lt;/code&gt; in this case.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# solve with ContinuousCallback (implicit event)
-condition2(u,t,integrator) = u[1] # Event happens when condition2(u,t,integrator) == 0
-cb2 = ContinuousCallback(condition2,affect!,save_positions=(true,true))
-sol2 = solve(prob,Tsit5(),callback=cb2,saveat=0.1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# plot forward trajectory
-pl1 = plot(sol1, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;explicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-pl2 = plot(sol2, label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], title=&amp;quot;implicit event&amp;quot;, labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright)
-hline!(pl1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-hline!(pl2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-pl = plot(pl1,pl2)
-savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# solve with ContinuousCallback (implicit event)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Event happens when condition2(u,t,integrator) == 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ContinuousCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;condition2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can verify that both callbacks lead to the same forward time evolution (for fixed initial conditions and parameters).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;explicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;title&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;implicit event&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;BB_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -208,18 +212,14 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB_forward.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/BB_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In addition, the implicitly defined impact time via the &lt;code&gt;ContinuousCallback&lt;/code&gt; also changes appropriately when changing the initial conditions or the parameters, for example when using  $\mathrm g = 9.81$ for the gravitation constant. In other words, the event time $\tau=\tau(p,z_0,v_0,t_0)$ is a function of the parameters and initial conditions, and is implicitly defined by the event condition.&lt;/p&gt;
 &lt;p&gt;Suppose we let the ball drop from a somewhat higher position now. Does an increase in height $z$ at $t=0$ give an increase or decrease in height at the end time $t_\text{end}=1.9$? This is something we can answer with sensitivity analysis. For example if we increase the height by (a fraction of) one unit then using $(\star)$&lt;/p&gt;
@@ -228,36 +228,35 @@ savefig(pl,&amp;quot;BB_forward.png&amp;quot;)
 $$&lt;/p&gt;
 &lt;p&gt;meaning the height at $t_\text{end}$ is also by a corresponding fraction of 0.84 units higher.&lt;/p&gt;
 &lt;p&gt;We can verify this visually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# animate forward trajectory
-sol3 = solve(remake(prob,u0=[u0[1]+0.5,u0[2]]),Tsit5(),callback=cb2,saveat=0.01)
-
-plt2 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;, xlims=(t0,tend))
-hline!(plt2, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-plot!(plt2, sol3, tspan=(t0,tend), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)
-
-list_plots = []
-for t in sol3.t
-  tstart = 0.0
-
-  plt1 = plot(sol2, label = false, labelfontsize=20, legendfontsize=20, lw = 1, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, color=&amp;quot;black&amp;quot;)
-  hline!(plt1, [0.0], label=false, color=&amp;quot;black&amp;quot;)
-  plot!(plt1, sol3, tspan=(t0,t), color=[1 2], label = [&amp;quot;z(t)&amp;quot; &amp;quot;v(t)&amp;quot;], labelfontsize=20, legendfontsize=20, lw = 2, xlabel = &amp;quot;t&amp;quot;, legend=:bottomright, denseplot=true, xlims=(t0,tend), ylims=(-11,9))
-  scatter!(plt1,[t,t], sol3(t), color=[1, 2], label=false)
-  plt = plot(plt1,plt2)
-  push!(list_plots, plt)
-end
-
-plot(list_plots[100])
-
-anim = animate(list_plots,every=1)
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# animate forward trajectory&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]]),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# scatter!(plt2, [t2,t2], sol3(t2), color=[1, 2], label=false)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;black&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomright&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;11&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;color&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -269,18 +268,11 @@ anim = animate(list_plots,every=1)
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/BB.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/BB.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;BB.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;The original curve is shown in black in the figure above.&lt;/p&gt;
 &lt;h2 id=&#34;sensitivity-analysis-with-events&#34;&gt;Sensitivity analysis with events&lt;/h2&gt;
@@ -295,7 +287,7 @@ $$&lt;/p&gt;
 &lt;p&gt;Inserting our results for $z(t) = z_{\rm exp}(t)$ instead of $z(t) = z_{\rm imp}(t)$ at $t_{\text{end}}$, a different value for the sensitivity is obtained (a value ignoring the changes in $z$ due to changes in the bouncing time $\tau$), e.g.,&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L^{\text{exp}}}{\text{d} z_0} = \begin{cases}
-1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\&lt;br&gt;
+1 &amp;amp; \text{with } z(t) = z_{\rm exp}(t) ,\\
 0.84 &amp;amp; \text{with } z(t) = z_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -306,7 +298,7 @@ $$&lt;/p&gt;
 &lt;p&gt;we could be interested in the sensitivity of $L$ with respect $g$. Using the velocity $v(t)$ at the explicit time $t = 0.9905$ instead of the implicit $v(\tau-)$, again a different value for the sensitivity is obtained (a value again ignoring the changes in $\tau$ and not the one we are looking for), even though $\tau = 0.9905$ in both cases:&lt;/p&gt;
 &lt;p&gt;$$
 \quad \frac{\text{d}L}{\text{d} g} = \begin{cases}
--0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\&lt;br&gt;
+-0.99005 &amp;amp; \text{with } v(t) = v_{\rm exp}(t) ,\\
 -\frac{g}{\sqrt{v_0^2 + 2 g z_0}} = -0.99995 &amp;amp; \text{ with } v(t) = v_{\rm imp}(t) .
 \end{cases}
 $$&lt;/p&gt;
@@ -323,51 +315,48 @@ $$&lt;/p&gt;
 &lt;p&gt;We write&lt;/p&gt;
 &lt;p&gt;$$\text{solve}(t_0, x_0, t, p)$$&lt;/p&gt;
 &lt;p&gt;$(= x(t))$ for the functional solution of the ODE at time $t$.&lt;/p&gt;
-&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
+&lt;p&gt;Regarding the computation of the sensitivities (the derivatives of the function &lt;code&gt;solve&lt;/code&gt;), we may then choose one of the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available algorithms&lt;/a&gt; for the given differential equation. Currently, &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, &lt;code&gt;QuadratureAdjoint()&lt;/code&gt;, &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt;, &lt;code&gt;TrackerAdjoint()&lt;/code&gt;, and &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; are compatible with events in ordinary differential equations.&lt;/p&gt;
 &lt;p&gt;Let us focus on the &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; algorithm which computes the sensitivities&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\&lt;br&gt;
+\frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{0}} &amp;amp;= \lambda(t_{0}),\\
 \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}p} &amp;amp;= \lambda_{p}(t_{0}),
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;of a loss function $L$ acting on the final state with respect to the initial state and the parameters. It does so by solving an ODE for $\lambda(s)$ in reverse time from $t$ to $t_0$&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\&lt;br&gt;
+\frac{\text{d}\lambda(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(\rightarrow x(s), p, t)}{\text{d} x(s)} \\
 \frac{\text{d}\lambda_{p}(s)}{\text{d}s} &amp;amp;= -\lambda(s)^\dagger \frac{\text{d} f(x(s), \rightarrow p, s)}{\text{d} p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;with initial conditions:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\&lt;br&gt;
+\lambda(t)&amp;amp;= \frac{\text{d}\thinspace L(\text{solve}(t_0, x_0, t, p))}{\text{d}x_{T}}, \\
 \lambda_{p}(t) &amp;amp;= 0.
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The arrows ($\rightarrow$) indicate the variable with respect to which we differentiate, which will become important later when the same variable shows up in multiple function arguments.&lt;/p&gt;
-&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Note that computing the vector-Jacobian products (vjp) in the adjoint ODE requires the value of $x(s)$ along its trajectory. In &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, we recompute $x(s)$ &amp;ndash; together with the adjoint variables &amp;ndash; backwards in time starting with its final value $x(t)$. A derivation of the ODE adjoint is given in &lt;a href=&#34;https://mitmath.github.io/18337/lecture11/adjoints&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; MIT 18.337 lecture notes&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;&lt;code&gt;BacksolveAdjoint()&lt;/code&gt;, essentially the custom primitive differentiation rule of &lt;code&gt;solve&lt;/code&gt;, is the elementary building block needed to derive sensitivities also in more complicated examples:&lt;/p&gt;
 &lt;p&gt;Consider a loss depending on the state $x(s_i)$ at fix time points $s_i$ through loss functions $L_i^{\text{exp}}$,&lt;/p&gt;
 &lt;p&gt;$$
 L^{\text{exp}} = \sum_i L_i^{\text{exp}}(s_i, x(s_i), p).
 $$&lt;/p&gt;
 &lt;p&gt;Without contortions we can obtain the sensitivity of $L^{\text{exp}}$ in $p$ (or in $x_0$) using the tool we have. For those a bit familiar with automatic differentiation, this is perhaps easiest to see if we write $L^{\text{exp}}$ as pseudo code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1 = solve(t0, x0, s1, p)
-    L = L1(s1, x1, p)
-    x2 = solve(s1, x1, s2, p)
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and consider the problem of automatically differentiating it. You&amp;rsquo;ll just need the primitives of &lt;code&gt;solve&lt;/code&gt;, &lt;code&gt;L1&lt;/code&gt; and &lt;code&gt;L2&lt;/code&gt;! The sensitivities can be computed by repeated calls to &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; on the intervals &lt;code&gt;(s_i, s_{i+1})&lt;/code&gt; backward in time, taking in the sensitivities $\sum_i L^{\text{exp}}(s_i, x(s_i), p)$ at times $s_i$, or according to the common notation&lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; as single call &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; with discrete callbacks:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\&lt;br&gt;
+\frac{\text{d}\lambda(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(\rightarrow x(t), p, t)}{\text{d} x(t)} - \frac{\text{d} L_i^{\text{exp}}(\rightarrow x(t), p)}{\text{d} x(t)}^\dagger \delta(t-s_i), \\
 \frac{\text{d}\lambda_{p}(t)}{\text{d}t} &amp;amp;= -\lambda(t)^\dagger \frac{\text{d} f(x(t), \rightarrow p, t)}{\text{d} p} - \frac{\text{d} L_i^{\text{exp}}(  x(t),\rightarrow p)}{\text{d} p}^\dagger \delta(t-s_i).
 \end{aligned}
 $$&lt;/p&gt;
@@ -381,24 +370,23 @@ $$&lt;/p&gt;
 $$&lt;/p&gt;
 &lt;p&gt;Now we can even properly define the &lt;code&gt;rrule&lt;/code&gt; of &lt;code&gt;solve&lt;/code&gt; in the sense of &lt;code&gt;DiffRules.jl&lt;/code&gt;.&lt;/p&gt;
 &lt;h3 id=&#34;explicit-events-1&#34;&gt;Explicit events&lt;/h3&gt;
-&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss(t0, x0, p)
-    x1- = solve(t0, x0, s1, p)
-    L = L1-(s1, x1-, p) # saved before affect
-    x1 = a(x1-, p, s1) # affect
-    L += L1(s1, x1, p) # saved after affect
-    x2- = solve(s1, x1, s2, p)
-    L += L2-(s2, x2-, p)
-    x2 = a(x2-, p, s2) # affect
-    L += L2(s2, x2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
+&lt;p&gt;To make &lt;code&gt;BacksolveAdjoint()&lt;/code&gt; compatible with explicit events&lt;sup id=&#34;fnref2:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt;,&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved before affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# saved after affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# affect&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;s2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we have to store the event times $s_j$ as well as the state $x(s_j-)$ at the left limit of $s_i$.&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; We then solve the adjoint ODE backwards in time between the events. As soon as we reach an event time $s_j$ from the right, we update the augmented state according to&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\&lt;br&gt;
+\lambda({s_j}-) &amp;amp;= \lambda({s_j})^\dagger \frac{\text{d} a(\rightarrow x({s_j}-), p, {s_j}-)}{\text{d} x(s_j-)} \\
 \lambda_p({s_j}-) &amp;amp;= \lambda_p({s_j}) -  \lambda({s_j})^\dagger \frac{\text{d} a(x({s_j}-), \rightarrow p, {s_j}-)}{\text{d} p}
 \end{aligned}
 $$&lt;/p&gt;
@@ -412,27 +400,26 @@ $$&lt;/p&gt;
 &lt;p&gt;to put emphasis on this, or equivalently, compute for a unspecified function $L(t, x, p)$ the result of $\frac{\text{d}L}{\text{d}p}$ with&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\&lt;br&gt;
+\frac{\text{d}L}{\text{d}p} &amp;amp;= \frac{\text{d}L( \text{solve2}(t_0, x_0, g, \rightarrow p),\rightarrow p)}{\text{d}p}\\
 &amp;amp;= \frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;which indicates that changing $p$ influences $L$ both through changes in $\tau_1$ as well as changes in&lt;/p&gt;
 &lt;p&gt;$$\xi_1 = x(\tau_1-).$$&lt;/p&gt;
 &lt;p&gt;This case where we have a loss function $L = L_1$ depending on $\tau_1$, $x(\tau_1)$, and $p$
-was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+was also considered by Ricky T. Q. Chen, Brandon Amos, and Maximilian Nickel in their ICLR 2021 paper&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;p&gt;Therefore, the sensitivity of the event time with respect to parameters $\frac{\text{d}\tau}{\text{d}p}$ must be taken into account.
-Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the 
-&lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
+Here and in the following we consider only the $p$-dependence of $\tau_1$ for simplicity. However, it is straightforward to include a dependence on the initial state $x_0$ in an analogues way&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;In a first step, we need to compute the sensitivity of $\tau_1(p)$ with respect to $p$ (or $x_0$) based on the event condition $g(t, x(t)) = 0$.  We can apply the &lt;a href=&#34;https://www.uni-siegen.de/fb6/analysis/overhagen/vorlesungsbeschreibungen/skripte/analysis3_1.pdf&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implicit function theorem&lt;/a&gt;. For this, see that $\tau_1(p)$ is implicitly defined by $F(p, \tau_1) = g( \tau_1, \text{solve}(t_0, x_0, \tau_1, p)) = 0$ which yields&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\&lt;br&gt;
+\frac{\text{d}\tau_1(p)}{\text{d}p} &amp;amp;= - \left(\frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\right)^{-1} \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \tau_1, \rightarrow p))}{\text{d}p} .\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;The total derivative&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; inside the bracket is:
 $$
 \begin{aligned}
-\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\&lt;br&gt;
+\frac{\text{d}g}{\text{d}\tau_1} \stackrel{\text{def}}{=} \frac{\text{d}g(\rightarrow \tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1} &amp;amp;= \frac{\text{d}g(\rightarrow \tau_1, \xi_1)}{\text{d}\tau_1} + \frac{\text{d}g(\tau_1, \text{solve}(t_0, x_0, \rightarrow \tau_1, p))}{\text{d}\tau_1}\\
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;Since&lt;/p&gt;
@@ -455,8 +442,8 @@ for the second term of $\dfrac{\text{d}\tau_1(p)}{\text{d}p}$.&lt;/p&gt;
 &lt;p&gt;We can now write the gradient as:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\&lt;br&gt;
-+&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\&lt;br&gt;
+\frac{\text{d}L(\tau_1({\color{black}\rightarrow} p), \text{solve}(t_0, x_0, \tau_1({\color{black}\rightarrow}p), \rightarrow p),\rightarrow p)}{\text{d}p} &amp;amp;= \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  p), \rightarrow p)}{\text{d}p} \\
++&amp;amp; \frac{\text{d}L(\tau_1(p), \text{solve}(t_0, x_0, \tau_1(p),  \rightarrow p), p)}{\text{d}p} \\
 +&amp;amp; \frac{\text{d}L(\rightarrow \tau_1(p), \text{solve}(t_0, x_0, \rightarrow \tau_1(p),  p), p)}{\text{d}\tau_1} \frac{\text{d} \tau_1(p)}{\text{d}p},
 \end{aligned}
 $$&lt;/p&gt;
@@ -525,7 +512,7 @@ $$&lt;/p&gt;
 &lt;li&gt;If both limits contribute to the loss function, the contributions ${\color{red}\lambda_1}$ and ${\color{blue}\lambda_{a,1}}$ are added.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h4 id=&#34;generalization-several-events&#34;&gt;Generalization: several events&lt;/h4&gt;
-&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
+&lt;p&gt;As implied by Chen et al. as well as by Timo C. Wunderlich and Christian Pehle&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, one can chain together the events and differentiate through the entire time evolution on a time interval $(t_0, t_{\text{end}})$. That is, we are generally allowed to segment the time evolution over an interval $[t_0, t]$ into one from $[t_0, s]$ and a subsequent one from $[s, t]$:&lt;/p&gt;
 &lt;p&gt;$$
 \text{solve}(t_0, x_0, t, p)  = \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p),
 $$&lt;/p&gt;
@@ -533,16 +520,15 @@ $$&lt;/p&gt;
 &lt;p&gt;(A good exercise to get familiar with these type of arguments is to verify&lt;/p&gt;
 &lt;p&gt;$$\frac{\text{d}}{\text{d}s} \text{solve}(s, \text{solve}(t_0, x_0, s, p), t-s, p) = 0.)$$&lt;/p&gt;
 &lt;p&gt;Essentially, we know how to address several events already by considering a loss function&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function loss2(t0, x0, p)
-    tau1, xi1 = solve2(t0, x0, g, p)
-    L = L1(tau1, xi1, p)
-    tau2, xi2 = solve2(tau1, xi1, g, p)
-    L += L2(tau2, xi2, p)
-    ...
-    return L
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tau2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xi2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;L&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;and applying the same generic rules of (automatic) differentiation as in the previous example, now using the elementary differentiation rule for &lt;code&gt;solve2&lt;/code&gt; (&lt;code&gt;rrule&lt;/code&gt; for &lt;code&gt;solve2&lt;/code&gt;) we derived above.&lt;/p&gt;
 &lt;p&gt;Differentiating by hand, we have the following modification of the method from the previous section, which can be derived choosing a particular function $L$ in the previous section which incorporates the &lt;code&gt;solve&lt;/code&gt; from $\tau_1$ to $t_{\text{end}}$. Note&lt;/p&gt;
 &lt;p&gt;$$
 \lambda_{\tau_1}^0 = \frac{\text{d}(\text{solve}(\rightarrow \tau_1,  a(\xi_1,p), t_{\text{end}}, p))}{\text{d} \tau_1} = - f(a(\xi_1,p), p, \tau_1).
@@ -574,43 +560,38 @@ $$&lt;/p&gt;
 &lt;li&gt;Backpropagate $\lambda_J = {\color{red}\lambda_J} + {\color{blue}\lambda_{a,J}} + {\color{green}\lambda_{c,J}}$ to the next event time $\tau_{J-1}$ and iterate over the remaining $J-1$ events.&lt;/li&gt;
 &lt;/ol&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
-&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
-&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (
-&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;p&gt;We are still refining the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;). For further information, the interested reader is encouraged to track the associated issues &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/383&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#383&lt;/a&gt; and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/374&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;#374&lt;/a&gt;, and &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/445&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;PR #445&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;If you have any questions or comments, please don’t hesitate to contact us (&lt;a href=&#34;https://github.com/frankschae&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;github.com/frankschae&lt;/a&gt;)!&lt;/p&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Timo C. Wunderlich and Christian Pehle, Sci. Rep. &lt;em&gt;11&lt;/em&gt;, 12829 (2021).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$. &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;If the affect function also changes the parameters of the differential equation, we must additionally store $p(t_i-)$ and compute another vjp to update $\lambda_p$.&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;For a function $f$ of more than one variable $y = f(t, x_1(t),x_2(t),\dots,x_N(t))$, the &lt;a href=&#34;https://en.wikipedia.org/wiki/Differential_of_a_function#Differentials_in_several_variables&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;total derivative&lt;/a&gt; with respect to the independent variable $t$ is given by the sum of all partial derivatives
 $$
 \begin{aligned}
-\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\&lt;br&gt;
-&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\&lt;br&gt;
+\frac{\text{d}y}{\text{d}t} &amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(\rightarrow t),x_2(\rightarrow t),\dots,x_N(\rightarrow t))}{\text{d}t} \\
+&amp;amp;= \frac{\text{d}f(\rightarrow t, x_1(t),x_2(t),\dots,x_N(t))}{\text{d}t} + \frac{\text{d}f(t, x_1(\rightarrow t),x_2(t),\dots,x_N(t))}{\text{d}t}\\
 &amp;amp;+ \frac{\text{d}f(t, x_1(t),x_2(\rightarrow t),\dots,x_N(t))}{\text{d}t} + \dots +  \frac{\text{d}f(t, x_1(t),x_2(t),\dots,x_N(\rightarrow t))}{\text{d}t}.
 \end{aligned}
-$$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+$$&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -619,21 +600,11 @@ $$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-
       <link>https://frankschae.github.io/post/hybridde/</link>
       <pubDate>Wed, 16 Jun 2021 14:50:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/hybridde/</guid>
-      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
+      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
 &lt;p&gt;This first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).&lt;/p&gt;
-&lt;p&gt;Note: Please check the 
-&lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
+&lt;p&gt;Note: Please check the &lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
 &lt;h2 id=&#34;hybrid-differential-equations&#34;&gt;Hybrid Differential Equations&lt;/h2&gt;
-&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
+&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
 &lt;p&gt;Some relevant examples for hybrid DEs with discrete or continuous callbacks are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; (&lt;code&gt;ContinuousCallback&lt;/code&gt;).&lt;/li&gt;
@@ -643,7 +614,7 @@ $$ &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-
 &lt;li&gt;pharmacokinetic models&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, where explicit dosing times change the drug concentration in the blood (&lt;code&gt;DiscreteCallback&lt;/code&gt;). The simplest possible example being the one-compartment model.&lt;/li&gt;
 &lt;li&gt;kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (&lt;code&gt;DiscreteCallback&lt;/code&gt;).&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;kicked-harmonic-oscillator&#34;&gt;Kicked harmonic oscillator&lt;/h2&gt;
 &lt;p&gt;Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form&lt;/p&gt;
 &lt;p&gt;$$
@@ -652,40 +623,32 @@ $$&lt;/p&gt;
 &lt;p&gt;where $a=0.1$ and $b=1$ with initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-x(t=0) &amp;amp;= 1  \\&lt;br&gt;
+x(t=0) &amp;amp;= 1  \\
 v(t=0) &amp;amp;= \dot{x}(t=0) = 0.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;This second-order ODE can be 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use 
-&lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity
-using Zygote
-using Random
-u0 = Float32[1.; 0.]
-
-tspan = (0.0f0,50.0f0)
-
-dtsave = 0.5f0
-t = tspan[1]:dtsave:tspan[2]
-
-function oscillator!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  return nothing
-end
-
-prob_data = ODEProblem(oscillator!,u0,tspan)
-
-# ODE without kicks
-pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
+&lt;p&gt;This second-order ODE can be &lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use &lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqFlux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Optim&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# ODE without kicks&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -697,44 +660,36 @@ pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &am
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as &lt;code&gt;1&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;kicktimes = tspan[1]:1:tspan[2]
-function kick!(integrator)
-  integrator.u[end] += one(eltype(integrator.u))
-end
-cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false))
-
-sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t)
-t_data = sol_data.t
-ode_data = Array(sol_data)
-
-# visualize data
-pl1 = plot(t_data,ode_data[1,:],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl1,t_data,ode_data[2,:],label=&amp;quot;data v(t)&amp;quot;)
-
-pl2 = plot(t_data[1:20],ode_data[1,1:20],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl2,t_data[1:20],ode_data[2,1:20],label=&amp;quot;data v(t)&amp;quot;)
-pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;
-
-
-
-
-  
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualize data&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
 
 
@@ -746,63 +701,56 @@ pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting &lt;code&gt;save_positions=(true,true)&lt;/code&gt;, the kicks would be saved before &lt;strong&gt;and&lt;/strong&gt; after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;, we now aim at learning (potentially) missing parts of the model from these data traces.&lt;/p&gt;
 &lt;h3 id=&#34;high-domain-knowledge&#34;&gt;High domain knowledge&lt;/h3&gt;
 &lt;p&gt;For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the &lt;code&gt;affect!&lt;/code&gt; function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of &lt;code&gt;integrator.u&lt;/code&gt;. If we further set the initial value of that component to &lt;code&gt;1&lt;/code&gt;, then the neural network only has to learn that &lt;code&gt;du[3]&lt;/code&gt; is &lt;code&gt;0&lt;/code&gt;. In other words, the output of the neural network must be &lt;code&gt;0&lt;/code&gt; for all states &lt;code&gt;u&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;Random.seed!(123)
-nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1))
-p_nn1 = initial_params(nn1)
-
-function f1!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3] = nn1(u[1:2], p)[1]
-  return nothing
-end
-
-affect!(integrator) = integrator.u[2] += integrator.u[3]
-cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false))
-z0 = Float32[u0;one(u0[1])]
-prob1 = ODEProblem(f1!,z0,tspan,p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# to visualize the predictions of the trained neural network below
-function visualize(prob,p)
-  _prob = remake(prob,p=p)
-  ode_pred = Array(solve(_prob,Tsit5(),callback=cb,
-                 saveat=dtsave))[1:2,:]
-  pl1 = plot(t_data,ode_pred[1,:],label=&amp;quot;x(t)&amp;quot;)
-  scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=&amp;quot;data x(t)&amp;quot;)
-  pl2 = plot(t_data,ode_pred[2,:],label=&amp;quot;v(t)&amp;quot;)
-  scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=&amp;quot;data v(t)&amp;quot;)
-
-  pl = plot(pl1, pl2, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-  return pl, sum(abs2,ode_data .- ode_pred)
-end
-
-pl = plot(solve(prob1,Tsit5(),saveat=t,
-  callback=cb
-  ),label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;123&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FastChain&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tanh&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial_params&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# to visualize the predictions of the trained neural network below&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                 &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -814,76 +762,66 @@ pl = plot(solve(prob1,Tsit5(),saveat=t,
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;untrained_nn.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;which (of course) doesn&amp;rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### loss function
-function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint())
-  _prob = remake(prob,p=p)
-  pred = Array(solve(_prob,Tsit5(),callback=cb,
-               saveat=dtsave,sensealg=sensealg))[1:2,:]
-  sum(abs2,ode_data .- pred)
-end
-
-loss(p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The recently implemented tools are deeply hidden within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., 
-&lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;###################################
-# training loop
-# optimize the parameters for a few epochs with ADAM
-function train(prob, p_nn; sensealg=BacksolveAdjoint())
-  opt = ADAM(0.0003f0)
-  list_plots = []
-  losses = []
-  for epoch in 1:200
-    println(&amp;quot;epoch: $epoch / 200&amp;quot;)
-    _dy, back = Zygote.pullback(p -&amp;gt; loss(p,
-      prob=prob,
-      sensealg=sensealg), p_nn)
-    gs = @time back(one(_dy))[1]
-    push!(losses, _dy)
-    if epoch % 10 == 0
-      # plot every xth epoch
-      pl, test_loss = visualize(prob, p_nn)
-      println(&amp;quot;Loss (epoch: $epoch): $test_loss&amp;quot;)
-      display(pl)
-      push!(list_plots, pl)
-    end
-    Flux.Optimise.update!(opt, p_nn, gs)
-    println(&amp;quot;&amp;quot;)
-  end
-  return losses, list_plots
-end
-
-# plot training loss
-losses, list_plots = train(prob1, p_nn1)
-pl1 = plot(losses, lw = 1.5, xlabel = &amp;quot;epoch&amp;quot;, ylabel=&amp;quot;loss&amp;quot;, legend=false)
-pl2 = list_plots[end]
-pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t,
-   callback=cb
-  ), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-
-pl = plot(pl2,pl3)
-&lt;/code&gt;&lt;/pre&gt;
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### loss function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ReverseDiffAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;               &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The recently implemented tools are deeply hidden within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., &lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;###################################&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# training loop&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# optimize the parameters for a few epochs with ADAM&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ADAM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0003f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;200&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt; / 200&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;%&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;c&#34;&gt;# plot every xth epoch&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_loss&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Loss (epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;): &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$test_loss&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;display&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Optimise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;update!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot training loss&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;epoch&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;loss&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
-  
 
 
 
@@ -895,98 +833,87 @@ pl = plot(pl2,pl3)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/trained1.png&#34; &gt;
 
-
-  &lt;img src=&#34;https://frankschae.github.io/img/trained1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;trained1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We see the expected constant value of &lt;code&gt;u[3]&lt;/code&gt;, indicating a kick to the velocity of &lt;code&gt;+=1&lt;/code&gt;, at the kicking times over the full time interval.&lt;/p&gt;
 &lt;h2 id=&#34;reducing-the-domain-knowledge&#34;&gt;Reducing the domain knowledge&lt;/h2&gt;
-&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
+&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;changing the initial condition of the third component of &lt;code&gt;u&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;using another affect function &lt;code&gt;affect!(integrator) = integrator.u[2] = integrator.u[3]&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;dropping the knowledge that only &lt;code&gt;u[2]&lt;/code&gt; gets a kick by using a neural network with &lt;code&gt;2&lt;/code&gt; outputs (+ a fourth component in the ODE):&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4]
-function f2!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;fitting the parameters $a$ and $b$ simultaneously:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f3!(du,u,p,t)
-  a = p[end-1]
-  b = p[end]
-  nn_weights = p[1:end-2]
-
-  du[1] = u[2]
-  du[2] = -b*u[1] - a*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f3!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;inferring the entire underlying dynamics using a neural network with &lt;code&gt;4&lt;/code&gt; outputs:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f4!(du,u,p,t)
-  Ω = nn3(u[1:2], p)
-
-  du[1] = Ω[1]
-  du[2] = Ω[2]
-  du[3:4] .= Ω[3:4]
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f4!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;etc.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;refine the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;) and&lt;/li&gt;
-&lt;li&gt;support direct usage through the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
+&lt;li&gt;support direct usage through the &lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021) &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021)&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
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+      <div class="article-style">
+        Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris&rsquo; article3 for a detailed examination.
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+      <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
+    </div>
+
+    
+    <a href="/post/shadowing/"  class="summary-link">
+      <div class="article-style">
+        In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
+      </div>
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+      <a href="/post/hybridde/" >Neural Hybrid Differential Equations</a>
+    </div>
+
+    
+    <a href="/post/hybridde/"  class="summary-link">
+      <div class="article-style">
+        I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
+      </div>
+    </a>
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+      <a href="/post/high-weak/" >High weak order SDE solvers</a>
+    </div>
+
+    
+    <a href="/post/high-weak/"  class="summary-link">
+      <div class="article-style">
+        This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
+After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
+      </div>
+    </a>
+    
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+      <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a>
+    </div>
+
+    
+    <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/"  class="summary-link">
+      <div class="article-style">
+        First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
+      </div>
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-      <div class="article-style">
-        
-          Differentiable programming (∂P), i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning1. The Julia2 language is well suited for ∂P, see also Chris' article3 for a detailed examination.
-        
-      </div>
-    </div>
   
-    
-    
-    
-    <div>
-      <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></h2>
-      <div class="article-style">
-        
-          In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
-        
-      </div>
-    </div>
+
   
-    
-    
-    
-    <div>
-      <h2><a href="/post/hybridde/" >Neural Hybrid Differential Equations</a></h2>
-      <div class="article-style">
-        
-          I am delighted that I have been awarded my second GSoC stipend this year. I look forward to carrying out the ambitious project scope with my mentors Chris Rackauckas, Moritz Schauer, Yingbo Ma, and Mohamed Tarek.
-        
-      </div>
-    </div>
   
     
-    
-    
-    <div>
-      <h2><a href="/post/high-weak/" >High weak order SDE solvers</a></h2>
-      <div class="article-style">
-        
-          This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
-After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
-        
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+  
   
     
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-      <div class="article-style">
-        
-          First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
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-    <description>julia</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Sun, 01 Aug 2021 12:03:17 +0200</lastBuildDate>
+    <description>Julia</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sun, 01 Aug 2021 12:03:17 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
-      <title>julia</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Julia</title>
       <link>https://frankschae.github.io/tag/julia/</link>
     </image>
     
@@ -17,33 +17,17 @@
       <link>https://frankschae.github.io/post/abstract_differentiation/</link>
       <pubDate>Sun, 01 Aug 2021 12:03:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/abstract_differentiation/</guid>
-      <description>&lt;p&gt;
-&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is 
-&lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&#39; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and 
-&lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
-&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, 
-&lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different 
-&lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by 
-&lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a 
-&lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
-&lt;p&gt;
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s 
-&lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
+      <description>&lt;p&gt;&lt;a href=&#34;https://sinews.siam.org/Details-Page/scientific-machine-learning-how-julia-employs-differentiable-programming-to-do-it-best&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Differentiable programming (∂P)&lt;/a&gt;, i.e., the ability to differentiate general computer program structures, has enabled the efficient combination of existing packages for scientific computation and machine learning&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;. The Julia&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; language is &lt;a href=&#34;https://github.com/tensorflow/swift/blob/main/docs/WhySwiftForTensorFlow.md&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;well suited for ∂P&lt;/a&gt;, see also Chris&amp;rsquo; article&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt; for a detailed examination. There is already a plethora of examples where ∂P has provided massive performance &lt;em&gt;and&lt;/em&gt; accuracy advantages over black-box approaches to machine learning. This is because black-box machine learning approaches are flexible but require a large amount of data. Incorporating previously acquired knowledge about the structure of a problem reduces the amount of data and allows the learning task to be simplified&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, for example, by focusing on learning only the parts of the model that are actually missing&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;. In the context of quantum control, we have demonstrated the power of this framework for closed&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt; and &lt;a href=&#34;https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=3&amp;amp;t=12s&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;open quantum systems&lt;/a&gt;&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;∂P is (commonly) realized by automatic differentiation (AD), which is a family of techniques to efficiently and accurately differentiate numeric functions expressed as computer programs. Generally, besides forward- and reverse-mode AD, the two main branches of AD, &lt;a href=&#34;https://juliadiff.org/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;a large variety of software implementations&lt;/a&gt; with different &lt;a href=&#34;https://discourse.julialang.org/t/state-of-automatic-differentiation-in-julia/43083&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;pros and cons&lt;/a&gt; exists. The goal is to make the best choice in every part of the program without requiring users to significantly customize their code. Having a common ground by &lt;a href=&#34;https://github.com/JuliaDiff/ChainRules.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ChainRules.jl&lt;/a&gt; empowers this idea of a &lt;a href=&#34;http://www.stochasticlifestyle.com/glue-ad-for-full-language-differentiable-programming/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Glue AD&lt;/a&gt; where backend developers just define ChainRules overloads. However, switching from one backend to another on the user side can still be tedious because the user has to look up the syntax of the new AD package.&lt;/p&gt;
+&lt;p&gt;&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt; has started to &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implement a high level API for differentiation&lt;/a&gt; that unifies the APIs of all the AD packages in the Julia ecosystem.  Ultimately, the API of our new package, &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;AbstractDifferentiation.jl&lt;/a&gt;, aims at enabling AD users to write AD backend-agnostic code. This will greatly facilitate the switching between different AD packages. Once the interface is completed and all tests are added, it is also planned that &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; software suite adopts AbstractDifferentiation.jl as a better way of handling AD choices. In this part of my GSoC project, I&amp;rsquo;ve started to fix remaining errors of the &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;initial PR&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;The interested reader is encouraged to look at Mohamed&amp;rsquo;s &lt;a href=&#34;https://github.com/JuliaDiff/AbstractDifferentiation.jl/pull/1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first PR&lt;/a&gt; for a complete list of functions provided by AbstractDifferentiation.jl (and some great discussions about the package). In the rest of this blog post, I will focus on a concrete example to illustrate the main idea.&lt;/p&gt;
 &lt;h2 id=&#34;optimization-of-the-rosenbrock-function&#34;&gt;Optimization of the Rosenbrock function&lt;/h2&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
+&lt;p&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Rosenbrock_function&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Rosenbrock function&lt;/a&gt; is defined by&lt;/p&gt;
 &lt;p&gt;$$
 g(x_1,x_2) = (a-x_1)^2 + b(x_2-x_1^2)^2.
 $$&lt;/p&gt;
 &lt;p&gt;The function $g$ has a global minimum at $(x_1^\star, x_2^\star)= (a, a^2)$ with $g(x_1^\star, x_2^\star)=0$. In the following, we fix $a = 1$ and $b = 100$. The global minimum is located inside a long, narrow, banana-shaped, flat valley, which makes the function a common test case for optimization algorithms.&lt;/p&gt;
-&lt;p&gt;Let us now implement the 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
+&lt;p&gt;Let us now implement the &lt;a href=&#34;https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Gauss–Newton algorithm&lt;/a&gt; to find the global minimum. The Gauss–Newton algorithm iteratively finds the value of the $N$ variables ${\bf{x}}=(x_1,\dots, x_N)$ that minimize the sum of squares of $M$ residuals $(f_1,\dots, f_M)$&lt;/p&gt;
 &lt;p&gt;$$
 S({\bf x}) = \frac{1}{2} \sum_{i=1}^M f_i({\bf x})^2.
 $$&lt;/p&gt;
@@ -51,298 +35,275 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 {\bf x}^{k+1} = {\bf x}^k - \alpha_k \left(J^T J \right)^{-1} J^T f({\bf x}^k),
 $$
-where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a 
-&lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
+where $J$ is the Jacobian matrix at ${\bf{x}}^k$ and $\alpha_k$ is the step length determined via a &lt;a href=&#34;https://de.wikipedia.org/wiki/Gau%C3%9F-Newton-Verfahren#Beispiel&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;line search subroutine&lt;/a&gt;.&lt;/p&gt;
 &lt;p&gt;The following plot shows the Rosenbrock function in 3D as well as a 2D heatmap including the global minimum ${\bf x^\star}=(1,1)$ and our initial guess ${\bf x_0}=(0,-0.1)$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Pkg
-path = @__DIR__
-cd(path); Pkg.activate(&amp;quot;.&amp;quot;); Pkg.instantiate()
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@__DIR__&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;path&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;activate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;.&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Pkg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;instantiate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## AbstractDifferentiation is not released yet!!&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AbstractDifferentiation&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Enzyme&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;UnPack&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Rosenbrock function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualization&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;do_plot&lt;/span&gt;    
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.6&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.5&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;surface&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;linealpha&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;colorbar&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;camera&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;heatmap&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₁&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₂&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;c&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cgrad&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:thermal&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scale&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_1&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_2&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x_0&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:circle&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:green&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xopt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x^\star&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legendfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;15&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markershape&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:star&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markersize&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;markercolor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:red&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Rosenbrock.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-## AbstractDifferentiation is not released yet!!
 
-using AbstractDifferentiation
-using Test, LinearAlgebra
-using FiniteDifferences, ForwardDiff, Zygote
-using Enzyme, UnPack
-using Plots, LaTeXStrings
-# using Diffractor: ∂⃖¹ ## Diffractor needs &amp;gt;julia@1.6
 
-## Rosenbrock function
-# R: R^2 -&amp;gt; R: x -&amp;gt; (a-x₁)² + b(x₂-x₁²)²
-g(x,p) = (p[1]-x[1])^2 + p[2]*(x[2]-x[1]^2)^2
 
-# visualization
-p = [1.0,100.0]
-x₀ = [0.0,-0.1]
-xopt = [1.0,1.0]
 
-do_plot = true
-if do_plot    
-    x₁, x₂ = -2.0:0.01:2.0, -0.6:0.01:3.5
-    z = Surface((x₁,x₂)-&amp;gt;g([x₁,x₂],p), x₁, x₂)
-    pl1 = surface(x₁,x₂,z, linealpha = 0.3, c=cgrad(:thermal, scale = :exp), colorbar=true,
-                labelfontsize=20,camera = (3,50),
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
 
-    pl2 = heatmap(x₁,x₂,z, c=cgrad(:thermal, scale = :exp),
-                labelfontsize=20,
-                xlabel = L&amp;quot;x_1&amp;quot;, ylabel = L&amp;quot;x_2&amp;quot;)
-    scatter!(pl2, [(x₀[1],x₀[2])], label=L&amp;quot;x_0&amp;quot;, legendfontsize=15, markershape = :circle, markersize = 10, markercolor = :green)
-    scatter!(pl2, [(xopt[1],xopt[2])],label=L&amp;quot;x^\star&amp;quot;, legendfontsize=15, markershape = :star, markersize = 10, markercolor = :red)
 
-    pl = plot(pl1,pl2, layout=(2,1))
-    savefig(pl, &amp;quot;Rosenbrock.png&amp;quot;)
-end
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
 
 
-
-
-
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Rosenbrock.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Rosenbrock.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;To apply the Gauss-Newton algorithm to the Rosenbrock function $g$, we first cast $g$ into an appropriate form fulfilling $S({\bf x})$, i.e., we use:&lt;/p&gt;
 &lt;p&gt;$$
 f:\mathbb{R}^2\rightarrow\mathbb{R}^2:  {\bf x} \mapsto \begin{pmatrix}
-f_1({\bf x}) \\&lt;br&gt;
-f_2({\bf x}) \\&lt;br&gt;
+f_1({\bf x}) \\
+f_2({\bf x}) \\
 \end{pmatrix} = \begin{pmatrix}
-\sqrt{2}(a-x_1) \\&lt;br&gt;
-\sqrt{2b}(x_2-x_1^2)\\&lt;br&gt;
+\sqrt{2}(a-x_1) \\
+\sqrt{2b}(x_2-x_1^2)\\
 \end{pmatrix},
 $$&lt;/p&gt;
 &lt;p&gt;instead of $g$. We can easily compute the Jacobian of $f$ manually&lt;/p&gt;
 &lt;p&gt;$$
 J =  \begin{pmatrix}
--\sqrt{2} &amp;amp; 0 \\&lt;br&gt;
--2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\&lt;br&gt;
+-\sqrt{2} &amp;amp; 0 \\
+-2x_1\sqrt{2b} &amp;amp; \sqrt{2b} \\
 \end{pmatrix}.
 $$&lt;/p&gt;
 &lt;p&gt;We can then implement a (simple, non-optimized) version of the Gauss-Newton algorithm as follows.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# bring Rosenbrock function into the form &amp;quot;sum of squares of functions&amp;quot;
-f1(x,p) = convert(eltype(x),sqrt(2))*(p[1]-x[1])
-f2(x,p) = convert(eltype(x),sqrt(2*p[2]))*(x[2]-x[1]^2)
-f(x,p) = [f1(x,p),f2(x,p)]
-function f(res,x,p) # Enzyme works with inplace functions
-	res[1] = f1(x,p)
-	res[2] = f2(x,p)
-	return nothing
-end
-
-## manually pre-defined Jacobian
-function Jacobian(x,p)
-  [-convert(eltype(x),sqrt(2))   0
-  -2*x[1]*convert(eltype(x),sqrt(2*p[2]))  convert(eltype(x),sqrt(2*p[2]))]
-end
-
-## Gauss-Newton scheme
-function GaussNewton!(xs, x, p; maxiter=8, backend=nothing)
-    for i=1:maxiter
-        x = step(x, p, backend)
-        @info i
-        @show x
-        push!(xs, x)
-    end
-    return xs, x
-end
-done(x,x2,p) = g(x2,p) &amp;lt; g(x,p)
-function step(x, p, backend::Nothing, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = Jacobian(x,p)
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;xs = [x₀]
-GaussNewton!(xs, x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# output:
-[ Info: 1 ]
-x = [0.125, -0.08750000000000001]
-[ Info: 2 ]
-x = [0.234375, -0.047265625000000006]
-[ Info: 3 ]
-x = [0.4257812499999995, 0.06800537109374968]
-[ Info: 4 ]
-x = [0.5693359374999986, 0.21857223510742047]
-[ Info: 5 ]
-x = [0.784667968749996, 0.5165503501892037]
-[ Info: 6 ]
-x = [0.9999999999999961, 0.9536321163177449]
-[ Info: 7 ]
-x = [0.9999999999999989, 0.9999999999999999]
-[ Info: 8 ]
-x = [1.0, 1.0]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of 
-&lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## FiniteDifferences
-struct FDMBackend{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend() = FDMBackend(central_fdm(5, 1))
-const fdm_backend = FDMBackend()
-# Minimal interface
-AD.@primitive function jacobian(ab::FDMBackend, f, xs...)
-    return FiniteDifferences.jacobian(ab.alg, f, xs...)
-end
-
-# AD Jacobian returns tuple
-# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]
-# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]
-
-@test AD.jacobian(fdm_backend, x-&amp;gt;f(x,p), x₀)[1] ≈ Jacobian(x₀, p)
-@test AD.jacobian(fdm_backend, f, x₀, p)[1] ≈ Jacobian(x₀, p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function step(x, p, backend, α=1//1)
-  x2 = deepcopy(x)
-  while !done(x,x2,p)
-    J = AD.jacobian(backend, f, x, p)[1]
-    d = -inv(J&#39;*J)*J&#39;*f(x,p)
-    copyto!(x2,x + α*d)
-    α = α//2
-  end
-  return x2
-end
-
-
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=fdm_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If we want to use reverse-mode AD instead, for example via 
-&lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Zygote
-struct ZygoteBackend &amp;lt;: AD.AbstractReverseMode end
-const zygote_backend = ZygoteBackend()
-AD.@primitive function pullback_function(ab::ZygoteBackend, f, xs...)
-    return function (vs)
-        # Supports only single output
-        _, back = Zygote.pullback(f, xs...)
-        if vs isa AbstractVector
-            back(vs)
-        else
-            @assert length(vs) == 1
-            back(vs[1])
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(zygote_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=zygote_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (
-&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## ForwardDiff
-struct ForwardDiffBackend &amp;lt;: AD.AbstractForwardMode end
-const forwarddiff_backend = ForwardDiffBackend()
-AD.@primitive function pushforward_function(ab::ForwardDiffBackend, f, xs...)
-    # jvp = f&#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0
-    return function (vs)
-        if xs isa Tuple
-            @assert length(xs) &amp;lt;= 2
-            if length(xs) == 1
-                (ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1]),0),)
-            else
-                ForwardDiff.derivative(h-&amp;gt;f(xs[1]+h*vs[1], xs[2]+h*vs[2]),0)
-            end
-        else
-            ForwardDiff.derivative(h-&amp;gt;f(xs+h*vs),0)
-        end
-    end
-end
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(forwarddiff_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=forwarddiff_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;where we have used that the Jacobian-vector product $f&#39;(x)v$, i.e., the primitives of forward-mode AD, can be computed by 
-&lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
-&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. 
-&lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;## Enzyme
-struct EnzymeBackend{T1,T2,T3,T4} &amp;lt;: AD.AbstractReverseMode
-    out::T1
-    λ::T2
-    ∂f_∂x::T3
-    ∂f_∂p::T4
-end
-
-out = zero(x₀)
-λ = zero(x₀)
-∂f_∂x = zero(x₀)
-∂f_∂p = zero(p)
-
-const enzyme_backend = EnzymeBackend(out,λ,∂f_∂x,∂f_∂p)
-AD.@primitive function pullback_function(ab::EnzymeBackend, f, xs...)
-    return function (vs)  
-        # enzyme works only with inplace functions
-        if !(vs isa AbstractVector)
-            @assert length(vs) == 1 # Supports only single output
-            vs = vs[1]
-        end
-
-        if xs isa Tuple
-            @assert length(xs) == 2  # hard-coded for use case with two inputs
-            x₀ = xs[1]
-            p = xs[2]
-        end
-
-        @unpack out, λ, ∂f_∂x, ∂f_∂p = ab # cached in the struct, could also be created in here
-
-        ∂f_∂x .*= false
-        ∂f_∂p .*= false
-        out .*= false
-
-        copyto!(λ, vs)
-
-        autodiff(Duplicated(out, λ), Duplicated(x₀, ∂f_∂x), Duplicated(p, ∂f_∂p)) do _out,_x, _p
-            f(_out,_x,_p)
-        end
-        return (∂f_∂x,∂f_∂p)
-    end
-end
-AD.isinplace(ab::EnzymeBackend) = true
-AD.primalvalue(ab::EnzymeBackend, nothing, f, xs) = (f(ab.out,xs...);return ab.out)
-##
-
-@test minimum(AD.jacobian(fdm_backend, f, x₀, p) .≈ AD.jacobian(enzyme_backend, f, x₀, p))
-xs = [x₀]
-GaussNewton!(xs, x₀, p, backend=enzyme_backend)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# bring Rosenbrock function into the form &amp;#34;sum of squares of functions&amp;#34;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Enzyme works with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;	&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## manually pre-defined Jacobian&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;   &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Gauss-Newton scheme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;maxiter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;When we run the algorithm, we find the global minimum after about the 7th iteration.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# output:&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.125&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.08750000000000001&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.234375&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.047265625000000006&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.4257812499999995&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.06800537109374968&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5693359374999986&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.21857223510742047&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.784667968749996&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5165503501892037&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;6&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999961&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9536321163177449&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;7&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.9999999999999989&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.9999999999999999&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Info&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If computing the Jacobian by hand is too cumbersome (or not possible for other reasons), we can compute it using finite differences. Within the AbstractDifferentiation API, we can directly define, for instance, the Jacobian of &lt;a href=&#34;https://github.com/JuliaDiff/FiniteDifferences.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;FiniteDifferences.jl&lt;/a&gt; as a new primitive operation.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## FiniteDifferences&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Minimal interface&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FiniteDifferences&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# AD Jacobian returns tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dx = AD.jacobian(fdm_backend, f(x,p), x₀, p)[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# df_dp = AD.jacobian(fdm_backend, f(x,p), x₀, p)[2]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;After overloading the &lt;code&gt;step&lt;/code&gt; function, we can run the Gauss-Newton algorithm as follows:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;step&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;while&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;done&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;J&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;d&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;inv&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;J&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;d&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If we want to use reverse-mode AD instead, for example via &lt;a href=&#34;https://github.com/FluxML/Zygote.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Zygote.jl&lt;/a&gt;, a natural choice for the primitive is to define the pullback function. AbstractDifferentiation then generates the associated code to compute the Jacobian for us.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ZygoteBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Typically, reverse-mode AD is only beneficial for functions $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ where $M \ll N$, thus it is also a good idea to compare the performance with respect to forward-mode AD (&lt;a href=&#34;https://github.com/JuliaDiff/ForwardDiff.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ForwardDiff.jl&lt;/a&gt;)&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## ForwardDiff&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractForwardMode&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;ForwardDiffBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;c&#34;&gt;# jvp = f&amp;#39;(x)*v, i.e., differentiate f(x + h*v) wrt h at 0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;else&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;derivative&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;where we have used that the Jacobian-vector product $f&amp;rsquo;(x)v$, i.e., the primitives of forward-mode AD, can be computed by &lt;a href=&#34;https://discourse.julialang.org/t/help-with-jacobian-vector-product-to-get-natural-gradient/51115/12&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;differentiating $f(x + hv)$ with respect to $h$ at 0&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;Many AD packages, such as Zygote, have troubles with mutating functions. &lt;a href=&#34;https://github.com/wsmoses/Enzyme.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Enzyme.jl&lt;/a&gt; is one of the exceptions. Additionally, it is very fast and has further improved the performance of the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/pull/427#issuecomment-866509944&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;adjoints implemented within the DiffEqSensitivity package&lt;/a&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;## Enzyme&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractReverseMode&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;T4&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pullback_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;  
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;c&#34;&gt;# enzyme works only with inplace functions&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AbstractVector&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# Supports only single output&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;isa&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Tuple&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;nd&#34;&gt;@assert&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;  &lt;span class=&#34;c&#34;&gt;# hard-coded for use case with two inputs&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;nd&#34;&gt;@unpack&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# cached in the struct, could also be created in here&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;out&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.*=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;copyto!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;autodiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;λ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Duplicated&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;do&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;            &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;∂f_∂p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;isinplace&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;primalvalue&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;EnzymeBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;);&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;##&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;minimum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jacobian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;GaussNewton!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;backend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;enzyme_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have declared the Enzyme backend as &lt;code&gt;inplace&lt;/code&gt; (which is important for internal control flow) and specified a &lt;code&gt;primalvalue&lt;/code&gt; function returning the primal value of the forward pass.&lt;/p&gt;
 &lt;h2 id=&#34;some-current-glitches&#34;&gt;Some current glitches&lt;/h2&gt;
 &lt;p&gt;First, the push forward of a tuple of vectors, e.g., $(v_1, v_2)$, for a function with several input arguments is currently ambiguous. While &lt;code&gt;AD.jacobian&lt;/code&gt; primitives and &lt;code&gt;AD.pullback_function&lt;/code&gt; primitives interpret the push forward of our $f$ function as&lt;/p&gt;
 &lt;p&gt;$$
@@ -352,86 +313,77 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\partial f(x_0,p)}{\partial x} v_1 + \frac{\partial f(x_0,p)}{\partial p} v_2.
 $$&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous
-vs = (randn(2), randn(2))
-res1 = AD.pushforward_function(fdm_backend, f, x₀, p)(vs)
-res2 = AD.pushforward_function(forwarddiff_backend, f, x₀, p)(vs)
-
-@test res2 ≈ res1[1] + res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;res2a = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((vs[1], zero(vs[2])))
-res2b = AD.pushforward_function(forwarddiff_backend, f, x₀, p)((zero(vs[1]), vs[2]))
-
-@test res2a ≈ res1[1]
-@test res2b ≈ res1[2]
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;struct FDMBackend2{A} &amp;lt;: AD.AbstractFiniteDifference
-    alg::A
-end
-FDMBackend2() = FDMBackend2(central_fdm(5, 1))
-const fdm_backend2 = FDMBackend2()
-AD.@primitive function pushforward_function(ab::FDMBackend2, f, xs...)
-    return function (vs)
-        FDM.jvp(ab.alg, f, tuple.(xs, vs)...)
-    end
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian only defined with respect to single input variable
-@test_throws AssertionError H1 = AD.hessian(forwarddiff_backend, g, x₀, p)
-H1 = AD.hessian(forwarddiff_backend, x-&amp;gt;g(x,p), x₀)
-H2 = AD.hessian(forwarddiff_backend, p-&amp;gt;g(x₀,p), p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)
-backends = AD.HigherOrderBackend((forwarddiff_backend,zygote_backend))
-H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# pushforward_function wrt to multiple vectors is currently ambiguous&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;randn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fdm_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Thus, we currently solve this issue by augmenting the input in the case of &lt;code&gt;AD.pushforward_function&lt;/code&gt; primitives.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zero&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res2b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;≈&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The plural &amp;ldquo;primitives&amp;rdquo; is used here because we may have different &lt;code&gt;pushforward_function&lt;/code&gt; primitives for different backends. For instance, we can define an additional &lt;code&gt;pushforward_function&lt;/code&gt; primitive for FiniteDifferences by:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;struct&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;{&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;}&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;&amp;lt;:&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;AbstractFiniteDifference&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;A&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;central_fdm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;const&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;fdm_backend2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@primitive&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pushforward_function&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;::&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;FDMBackend2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;FDM&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;jvp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ab&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tuple&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;...&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Second, to avoid misunderstandings for the output of a Hessian of a function with several input arguments, we allow only single input arguments to the &lt;code&gt;Hessian&lt;/code&gt; function.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian only defined with respect to single input variable&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test_throws&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;AssertionError&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Third, computing the Hessian requires to nest AD/backend calls. This can lead to failure if one tries to use Zygote over Zygote. To solve this problem, we have implemented a &lt;code&gt;HigherOrderBackend&lt;/code&gt; that takes a tuple containing multiple backends (because, for example, using ForwardDiff over Zygote is perfectly fine).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Hessian might fail if AD system calls must not be nested (e.g. Zygote over Zygote)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;HigherOrderBackend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;forwarddiff_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zygote_backend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;H3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AD&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;hessian&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;backends&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;x&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;x₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;There are many other use cases, e.g.,&lt;/p&gt;
 &lt;ul&gt;
-&lt;li&gt;
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
-&lt;li&gt;The 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
+&lt;li&gt;&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Sensitivity analysis of differential equations&lt;/a&gt; requires vector-Jacobian products for adjoint methods and Jacobian-vector products for tangent methods.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton–Raphson method&lt;/a&gt; for rootfinding requires the gradient in the case of scalar function $f:\mathbb{R}\rightarrow\mathbb{R}$ and the Jacobian in case of $N$ (nonlinear) equations, i.e., finding the zeros of $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$.&lt;/li&gt;
+&lt;li&gt;The &lt;a href=&#34;https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Newton method&lt;/a&gt; in optimization requires the computation of the Hessian.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;AbstractDifferentiation.jl is by no means complete yet. We are still in the very early stages, but we hope to make significant progress in the coming weeks. Some of the next steps are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;fixing remaining bugs, e.g., with respect to the computation of the Hessian and&lt;/li&gt;
-&lt;li&gt;adding AD/Finite Differentiation packages such as 
-&lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
+&lt;li&gt;adding AD/Finite Differentiation packages such as &lt;a href=&#34;https://github.com/JuliaDiff/Diffractor.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Diffractor&lt;/a&gt;.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Mike Innes, Alan Edelman, et al., arXiv preprint arXiv:1907.07587 (2019).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Jeff Bezanson, Stefan Karpinski, et al., arXiv preprint arXiv:1209.5145 (2012).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, The Winnower 8, DOI: 10.15200/winn.156631.13064 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Raj Dandekar, Chris Rackauckas, et al., Patterns &lt;strong&gt;1&lt;/strong&gt;, 100145 (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -441,8 +393,7 @@ H3 = AD.hessian(backends, x-&amp;gt;g(x,p), x₀)
       <pubDate>Fri, 02 Jul 2021 11:08:22 +0200</pubDate>
       <guid>https://frankschae.github.io/post/shadowing/</guid>
       <description>&lt;p&gt;In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &amp;ldquo;butterfly effect&amp;rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.&lt;/p&gt;
-&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
+&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:&lt;/p&gt;
 &lt;p&gt;$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -456,57 +407,53 @@ Under the assumption of ergodicity, $\langle g \rangle_∞$ only depends on $p$.
 &lt;p&gt;One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{pmatrix}
-\text{d}x \\&lt;br&gt;
-\text{d}y \\&lt;br&gt;
-\text{d}z \\&lt;br&gt;
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\&lt;br&gt;
-x(\rho-z) - y\\&lt;br&gt;
-x y - \beta z \\&lt;br&gt;
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
-
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
-
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &amp;quot;initial&amp;quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&amp;quot;attractor&amp;quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Statistics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;QuadGK&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SparseArrays&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# simulate 1 trajectory of the Lorenz system forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;28.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;initial&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;attractor&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -518,79 +465,71 @@ savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.&lt;/p&gt;
-&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
+&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
 &lt;p&gt;$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function compute_objective(sol)
-  quadgk(t-&amp;gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&amp;quot;z(t)&amp;quot;, xlabel=L&amp;quot;t&amp;quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&amp;quot;\langle z\rangle&amp;quot;, lw = 2)
-savefig(pl2, &amp;quot;zsingle.png&amp;quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
-
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
-
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&amp;quot;\langle z\rangle&amp;quot;, xlabel=L&amp;quot;\rho&amp;quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &amp;quot;zvsrho.png&amp;quot;)
-
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &amp;quot;z.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We obtain:&lt;/p&gt;
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;quadgk&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;atol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;mean_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zsingle.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# for each value of the parameter, solve 20 times the initial value problem&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# wrap the procedure inside a function depending on p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ribbon&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;ho&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zvsrho.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;margin&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;600&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;300&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;z.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We obtain:&lt;/p&gt;
 
 
 
-  
 
 
 
@@ -602,40 +541,34 @@ savefig(pl4, &amp;quot;z.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/z.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/z.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;z.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$&lt;/p&gt;
 &lt;h2 id=&#34;conventional-forward-mode-sensitivity-analysis-and-finite-differencing&#34;&gt;Conventional forward-mode sensitivity analysis and finite-differencing&lt;/h2&gt;
-&lt;p&gt;For non-chaotic systems, we would just use the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we find diverging values:&lt;/p&gt;
+&lt;p&gt;For non-chaotic systems, we would just use the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_forward&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_calculus&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we find diverging values:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\&lt;br&gt;
+&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$&lt;/p&gt;
@@ -643,44 +576,39 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$&lt;/p&gt;
-&lt;p&gt;Similarly, using 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-     xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&amp;quot;Float32&amp;quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
-
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&amp;quot;Float32&amp;quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-
-savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;p&gt;Similarly, using &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;500&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_Floats.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -692,18 +620,14 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Without animation:&lt;/p&gt;
 
@@ -711,9 +635,6 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-  
-
-
 
 
 
@@ -723,60 +644,52 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_Floats.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;Luckily, the 
-&lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
+&lt;p&gt;Luckily, the &lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
 &lt;blockquote&gt;
 &lt;p&gt;Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&amp;ldquo;shadows&amp;rdquo;) the numerically computed one.&lt;/p&gt;
 &lt;/blockquote&gt;
 &lt;h2 id=&#34;shadowing-methods&#34;&gt;Shadowing methods&lt;/h2&gt;
-&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref2:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref3:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;lss-and-nilss-for-the-lorenz-system&#34;&gt;LSS and NILSS for the Lorenz system&lt;/h2&gt;
-&lt;p&gt;Switching to LSS or NILSS within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
+&lt;p&gt;Switching to LSS or NILSS within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# objective&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# LSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;CosWindowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0095888187322035&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Cos2Windowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0343951385924328&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0284286902740765&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_adjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.028428690274077&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# select via sensealg in solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9694728321500617&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;&lt;code&gt;CosWindowing()&lt;/code&gt; (default)&lt;/li&gt;
 &lt;li&gt;&lt;code&gt;Cos2Windowing()&lt;/code&gt;&lt;/li&gt;
@@ -785,72 +698,68 @@ dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
 &lt;p&gt;Additionally, an adjoint implementation &lt;code&gt;AdjointLSS()&lt;/code&gt; is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that &lt;code&gt;AdjointLSS(alpha=10.0)&lt;/code&gt; agrees perfectly with &lt;code&gt;ForwardLSS(alpha=10.0)&lt;/code&gt;. In all cases considered, we find the expected sensitivity value of $\approx 1$.&lt;/p&gt;
 &lt;p&gt;However, the use of &lt;code&gt;LSS()&lt;/code&gt; is (typically) much more expensive than the use of &lt;code&gt;NILSS()&lt;/code&gt;, because &lt;code&gt;LSS()&lt;/code&gt; needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of &lt;code&gt;NILSS()&lt;/code&gt; scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making &lt;code&gt;NILSS()&lt;/code&gt; more efficient.&lt;/p&gt;
 &lt;p&gt;In the &lt;code&gt;NILSS()&lt;/code&gt; algorithm, the user can control the number of steps per segment as well as the number of segments.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# NILSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# make sure trajectory is fully on the attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;120.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9966924374966089&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0416028730638789&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Warning: Detected a large value of ξ at the beginning of a segment.&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref4:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;NILSAS&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;li&gt;FD-NILSS&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref4:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019) &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019)&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019) &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019)&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -859,21 +768,11 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
       <link>https://frankschae.github.io/post/hybridde/</link>
       <pubDate>Wed, 16 Jun 2021 14:50:17 +0200</pubDate>
       <guid>https://frankschae.github.io/post/hybridde/</guid>
-      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and 
-&lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
+      <description>&lt;p&gt;I am delighted that I have been awarded my second GSoC stipend this year.  I look forward to carrying out the ambitious project scope with my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;,  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt;, and &lt;a href=&#34;https://github.com/mohamed82008&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Mohamed Tarek&lt;/a&gt;. This year&amp;rsquo;s project is embedded within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/5765643267211264/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;NumFocus&lt;/a&gt;/&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization and comprises adjoint sensitivity methods for discontinuities, shadowing methods for chaotic dynamics, symbolically generated adjoint methods, and further AD tooling within the Julia Language.&lt;/p&gt;
 &lt;p&gt;This first post aims to illustrate our new (adjoint) sensitivity analysis tools with respect to event handling in (ordinary) differential equations (DEs).&lt;/p&gt;
-&lt;p&gt;Note: Please check the 
-&lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
+&lt;p&gt;Note: Please check the &lt;a href=&#34;https://docs.sciml.ai/SciMLSensitivity/dev/examples/hybrid_jump/hybrid_diffeq/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciMLSensitivity.jl docs&lt;/a&gt; for a maintained neural hybrid DE tutorial!&lt;/p&gt;
 &lt;h2 id=&#34;hybrid-differential-equations&#34;&gt;Hybrid Differential Equations&lt;/h2&gt;
-&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
+&lt;p&gt;DEs with additional explicit or implicit discontinuities are called hybrid DEs. Within the SciML software suite, such discontinuities may be incorporated into DE models by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;callbacks&lt;/a&gt;. Evidently, the incorporation of discontinuities allows a user to specify changes (&lt;em&gt;events&lt;/em&gt;) in the system, i.e., changes of the state or the parameters of the DE, which cannot be modeled by a plain ordinary DE. While explicit events can be described by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#DiscreteCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiscreteCallbacks&lt;/a&gt;, implicit events have to be specified by &lt;a href=&#34;https://diffeq.sciml.ai/stable/features/callback_functions/#ContinuousCallback-Examples&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;ContinuousCallbacks&lt;/a&gt;. That is, explicit events possess explicit event times, while implicit events are triggered when a continuous function evaluates to &lt;code&gt;0&lt;/code&gt;. Thus, implicit events require some sort of rootfinding procedure.&lt;/p&gt;
 &lt;p&gt;Some relevant examples for hybrid DEs with discrete or continuous callbacks are:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;quantum optics experiments, where photon-counting measurements lead to jumps in the quantum state that occur with a variable rate, see for instance Appendix A in Ref.&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; (&lt;code&gt;ContinuousCallback&lt;/code&gt;).&lt;/li&gt;
@@ -883,7 +782,7 @@ nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
 &lt;li&gt;pharmacokinetic models&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, where explicit dosing times change the drug concentration in the blood (&lt;code&gt;DiscreteCallback&lt;/code&gt;). The simplest possible example being the one-compartment model.&lt;/li&gt;
 &lt;li&gt;kicked oscillator dynamics, e.g., a harmonic oscillator that gets a kick at some time points (&lt;code&gt;DiscreteCallback&lt;/code&gt;).&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The associated sensitivity methods that allow us to differentiate through the respective hybrid DE systems have been recently introduced in Refs. &lt;sup id=&#34;fnref1:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;kicked-harmonic-oscillator&#34;&gt;Kicked harmonic oscillator&lt;/h2&gt;
 &lt;p&gt;Let us consider the simple physical model of a damped harmonic oscillator, described by an ODE of the form&lt;/p&gt;
 &lt;p&gt;$$
@@ -892,40 +791,40 @@ $$&lt;/p&gt;
 &lt;p&gt;where $a=0.1$ and $b=1$ with initial conditions&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-x(t=0) &amp;amp;= 1  \\&lt;br&gt;
+x(t=0) &amp;amp;= 1  \\
 v(t=0) &amp;amp;= \dot{x}(t=0) = 0.
 \end{aligned}&lt;br&gt;
 $$&lt;/p&gt;
-&lt;p&gt;This second-order ODE can be 
-&lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use 
-&lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DiffEqFlux, DifferentialEquations, Flux, Optim, Plots, DiffEqSensitivity
-using Zygote
-using Random
-u0 = Float32[1.; 0.]
-
-tspan = (0.0f0,50.0f0)
-
-dtsave = 0.5f0
-t = tspan[1]:dtsave:tspan[2]
+&lt;p&gt;This second-order ODE can be &lt;a href=&#34;https://en.wikipedia.org/wiki/Ordinary_differential_equation#Reduction_of_order&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;reduced&lt;/a&gt; to two first-order ODEs, such that we can straightforwardly simulate the resulting ODE with the &lt;code&gt;DifferentialEquations.jl&lt;/code&gt; package. (Instead of doing this reduction manually, we could also use &lt;a href=&#34;https://mtk.sciml.ai/stable/tutorials/higher_order/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;&lt;code&gt;ModelingToolkit.jl&lt;/code&gt;&lt;/a&gt; to transform the ODE in an automatic manner. Alternatively, for second-order ODEs, there is also a &lt;code&gt;SecondOrderODEProblem&lt;/code&gt; implemented.) The Julia code reads:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqFlux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Optim&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;oscillator!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# ODE without kicks&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-function oscillator!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  return nothing
-end
 
-prob_data = ODEProblem(oscillator!,u0,tspan)
 
-# ODE without kicks
-pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -935,46 +834,36 @@ pl = plot(solve(prob_data,Tsit5(),saveat=t), label=[&amp;quot;x(t)&amp;quot; &am
 
 
 
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator_no_kicks.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We now include a kick to the velocity of the oscillator at regular time steps. Here, we choose both the time difference between the kicks and the increase in velocity as &lt;code&gt;1&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;kicktimes = tspan[1]:1:tspan[2]
-function kick!(integrator)
-  integrator.u[end] += one(eltype(integrator.u))
-end
-cb_ = PresetTimeCallback(kicktimes,kick!,save_positions=(false,false))
-
-sol_data = solve(prob_data,Tsit5(),callback=cb_,saveat=t)
-t_data = sol_data.t
-ode_data = Array(sol_data)
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;eltype&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kick!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb_&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# visualize data&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;
 
-# visualize data
-pl1 = plot(t_data,ode_data[1,:],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl1,t_data,ode_data[2,:],label=&amp;quot;data v(t)&amp;quot;)
 
-pl2 = plot(t_data[1:20],ode_data[1,1:20],label=&amp;quot;data x(t)&amp;quot;)
-plot!(pl2,t_data[1:20],ode_data[2,1:20],label=&amp;quot;data v(t)&amp;quot;)
-pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;
 
 
 
 
-  
 
 
 
@@ -986,63 +875,54 @@ pl = plot(pl2, pl1, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/forward_damped_oscillator.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;forward_damped_oscillator.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 The left-hand side shows a zoom for short times to better resolve the kicks. Note that by setting &lt;code&gt;save_positions=(true,true)&lt;/code&gt;, the kicks would be saved before &lt;strong&gt;and&lt;/strong&gt; after the event such that the kicks would appear completely vertically in the plot. The data on the right-hand will be used as training data below. In the spirit of universal differential equations&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;, we now aim at learning (potentially) missing parts of the model from these data traces.&lt;/p&gt;
 &lt;h3 id=&#34;high-domain-knowledge&#34;&gt;High domain knowledge&lt;/h3&gt;
 &lt;p&gt;For simplicity, we assume that we have almost perfect knowledge about our system. That is, we assume to know the basic structure of the ODE, including its parameters $a$ and $b$, and that the &lt;code&gt;affect!&lt;/code&gt; function of the event only acts on the velocity. We then encode the affect as an additional component to the ODE. The task is thus to learn the dynamics of the third component of &lt;code&gt;integrator.u&lt;/code&gt;. If we further set the initial value of that component to &lt;code&gt;1&lt;/code&gt;, then the neural network only has to learn that &lt;code&gt;du[3]&lt;/code&gt; is &lt;code&gt;0&lt;/code&gt;. In other words, the output of the neural network must be &lt;code&gt;0&lt;/code&gt; for all states &lt;code&gt;u&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;Random.seed!(123)
-nn1 = FastChain(FastDense(2, 64, tanh),FastDense(64, 1))
-p_nn1 = initial_params(nn1)
-
-function f1!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3] = nn1(u[1:2], p)[1]
-  return nothing
-end
-
-affect!(integrator) = integrator.u[2] += integrator.u[3]
-cb = PresetTimeCallback(kicktimes,affect!,save_positions=(false,false))
-z0 = Float32[u0;one(u0[1])]
-prob1 = ODEProblem(f1!,z0,tspan,p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# to visualize the predictions of the trained neural network below
-function visualize(prob,p)
-  _prob = remake(prob,p=p)
-  ode_pred = Array(solve(_prob,Tsit5(),callback=cb,
-                 saveat=dtsave))[1:2,:]
-  pl1 = plot(t_data,ode_pred[1,:],label=&amp;quot;x(t)&amp;quot;)
-  scatter!(pl1,t_data[1:5:end],ode_data[1,1:5:end],label=&amp;quot;data x(t)&amp;quot;)
-  pl2 = plot(t_data,ode_pred[2,:],label=&amp;quot;v(t)&amp;quot;)
-  scatter!(pl2,t_data[1:5:end],ode_data[2,1:5:end],label=&amp;quot;data v(t)&amp;quot;)
-
-  pl = plot(pl1, pl2, layout=(1,2), xlabel=&amp;quot;t&amp;quot;)
-  return pl, sum(abs2,ode_data .- ode_pred)
-end
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;123&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;FastChain&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tanh&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;FastDense&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;64&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;initial_params&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;+=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;PresetTimeCallback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;kicktimes&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;affect!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_positions&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We can easily compare the time evolution of the neural hybrid DE with respect to the data:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# to visualize the predictions of the trained neural network below&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                 &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data x(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;scatter!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;data v(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ode_pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-pl = plot(solve(prob1,Tsit5(),saveat=t,
-  callback=cb
-  ),label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -1054,76 +934,64 @@ pl = plot(solve(prob1,Tsit5(),saveat=t,
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/untrained_nn.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;untrained_nn.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;which (of course) doesn&amp;rsquo;t match the data due to the random initialization of the neural network parameters before training. The neural network can be trained, i.e., its parameters can be optimized, by minimizing a mean-squared error loss function:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;### loss function
-function loss(p; prob=prob1, sensealg = ReverseDiffAdjoint())
-  _prob = remake(prob,p=p)
-  pred = Array(solve(_prob,Tsit5(),callback=cb,
-               saveat=dtsave,sensealg=sensealg))[1:2,:]
-  sum(abs2,ode_data .- pred)
-end
-
-loss(p_nn1)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The recently implemented tools are deeply hidden within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., 
-&lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&#39; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;###################################
-# training loop
-# optimize the parameters for a few epochs with ADAM
-function train(prob, p_nn; sensealg=BacksolveAdjoint())
-  opt = ADAM(0.0003f0)
-  list_plots = []
-  losses = []
-  for epoch in 1:200
-    println(&amp;quot;epoch: $epoch / 200&amp;quot;)
-    _dy, back = Zygote.pullback(p -&amp;gt; loss(p,
-      prob=prob,
-      sensealg=sensealg), p_nn)
-    gs = @time back(one(_dy))[1]
-    push!(losses, _dy)
-    if epoch % 10 == 0
-      # plot every xth epoch
-      pl, test_loss = visualize(prob, p_nn)
-      println(&amp;quot;Loss (epoch: $epoch): $test_loss&amp;quot;)
-      display(pl)
-      push!(list_plots, pl)
-    end
-    Flux.Optimise.update!(opt, p_nn, gs)
-    println(&amp;quot;&amp;quot;)
-  end
-  return losses, list_plots
-end
-
-# plot training loss
-losses, list_plots = train(prob1, p_nn1)
-pl1 = plot(losses, lw = 1.5, xlabel = &amp;quot;epoch&amp;quot;, ylabel=&amp;quot;loss&amp;quot;, legend=false)
-pl2 = list_plots[end]
-pl3 = plot(solve(prob1,p=p_nn1,Tsit5(),saveat=t,
-   callback=cb
-  ), label=[&amp;quot;x(t)&amp;quot; &amp;quot;v(t)&amp;quot; &amp;quot;u3(t)&amp;quot;])
-
-pl = plot(pl2,pl3)
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;### loss function&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ReverseDiffAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Array&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;               &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dtsave&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abs2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ode_data&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pred&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The recently implemented tools are deeply hidden within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package. However, while the user could previously only choose discrete sensitivities such as &lt;code&gt;ReverseDiffAdjoint()&lt;/code&gt; or  &lt;code&gt;ForwardDiffAdjoint()&lt;/code&gt; that rely on direct differentiation through the solver operations to get accurate gradients, one can now also select continuous adjoint sensitivity methods such as &lt;code&gt;BacksolveAdjoint()&lt;/code&gt;,  &lt;code&gt;InterpolatingAdjoint()&lt;/code&gt;, and &lt;code&gt;QuadratureAdjoint()&lt;/code&gt; as the &lt;code&gt;sensealg&lt;/code&gt; for hybrid DEs. Each choice has its own characteristics in terms of stability, scaling with parameters, and memory consumption, see, e.g., &lt;a href=&#34;https://www.youtube.com/watch?v=XRJ-rtP2fVE&amp;amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;amp;index=1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris&amp;rsquo; talk&lt;/a&gt; at the SciML symposium at SIAM CSE.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;###################################&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# training loop&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# optimize the parameters for a few epochs with ADAM&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ADAM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0003f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;200&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt; / 200&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pullback&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;loss&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;back&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;one&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;_dy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;if&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;epoch&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;%&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;==&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;c&#34;&gt;# plot every xth epoch&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_loss&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;visualize&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Loss (epoch: &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$epoch&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;): &lt;/span&gt;&lt;span class=&#34;si&#34;&gt;$test_loss&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;display&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;      &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;Flux&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Optimise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;update!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;opt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;gs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;println&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# plot training loss&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;train&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;losses&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;epoch&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;loss&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p_nn1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;n&#34;&gt;callback&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;cb&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;v(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;u3(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -1135,98 +1003,85 @@ pl = plot(pl2,pl3)
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/trained1.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/trained1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;trained1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;We see the expected constant value of &lt;code&gt;u[3]&lt;/code&gt;, indicating a kick to the velocity of &lt;code&gt;+=1&lt;/code&gt;, at the kicking times over the full time interval.&lt;/p&gt;
 &lt;h2 id=&#34;reducing-the-domain-knowledge&#34;&gt;Reducing the domain knowledge&lt;/h2&gt;
-&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
+&lt;p&gt;If less physical information is included in the model design, the training becomes more difficult, e.g., due to &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/local_minima/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;local minima&lt;/a&gt;. Possible modification for the kicked oscillator could be&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;changing the initial condition of the third component of &lt;code&gt;u&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;using another affect function &lt;code&gt;affect!(integrator) = integrator.u[2] = integrator.u[3]&lt;/code&gt;,&lt;/li&gt;
 &lt;li&gt;dropping the knowledge that only &lt;code&gt;u[2]&lt;/code&gt; gets a kick by using a neural network with &lt;code&gt;2&lt;/code&gt; outputs (+ a fourth component in the ODE):&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;affect2!(integrator) = integrator.u[1:2] = integrator.u[3:4]
-function f2!(du,u,p,t)
-  du[1] = u[2]
-  du[2] = - u[1] - 1//10*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;affect2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;integrator&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f2!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;fitting the parameters $a$ and $b$ simultaneously:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f3!(du,u,p,t)
-  a = p[end-1]
-  b = p[end]
-  nn_weights = p[1:end-2]
-
-  du[1] = u[2]
-  du[2] = -b*u[1] - a*u[2]
-  du[3:4] .= nn2(u[1:2], nn_weights)
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f3!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn_weights&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;inferring the entire underlying dynamics using a neural network with &lt;code&gt;4&lt;/code&gt; outputs:&lt;/li&gt;
 &lt;/ul&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f4!(du,u,p,t)
-  Ω = nn3(u[1:2], p)
-
-  du[1] = Ω[1]
-  du[2] = Ω[2]
-  du[3:4] .= Ω[3:4]
-  return nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;ul&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f4!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nn3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Ω&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;ul&gt;
 &lt;li&gt;etc.&lt;/li&gt;
 &lt;/ul&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the adjoint sensitivity methods for hybrid DEs, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;refine the adjoints in case of implicit discontinuities (&lt;code&gt;ContinuousCallbacks&lt;/code&gt;) and&lt;/li&gt;
-&lt;li&gt;support direct usage through the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
+&lt;li&gt;support direct usage through the &lt;a href=&#34;https://diffeq.sciml.ai/stable/types/jump_types/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;jump problem&lt;/a&gt; interface&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information, the interested reader is encouraged to look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021) &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021)&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Ricky T. Q. Chen, Brandon Amos, Maximilian Nickel, arXiv preprint arXiv:2011.03902 (2020).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Junteng Jia, Austin R. Benson, arXiv preprint arXiv:1905.10403 (2019).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Michael Poli, Stefano Massaroli, et al., arXiv preprint arXiv:2106.04165 (2021).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., &amp;ldquo;Accelerated predictive healthcare analytics with pumas, a high performance pharmaceutical modeling and simulation platform.&amp;rdquo; (2020).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1235,11 +1090,8 @@ end
       <link>https://frankschae.github.io/post/high-weak/</link>
       <pubDate>Mon, 17 Aug 2020 14:46:35 +0200</pubDate>
       <guid>https://frankschae.github.io/post/high-weak/</guid>
-      <description>&lt;p&gt;This post summarizes our new 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
-for the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
+      <description>&lt;p&gt;This post summarizes our new &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
+for the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
 &lt;p&gt;After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
 Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics.
 Finally, we discuss how to run simulations on GPU hardware.&lt;/p&gt;
@@ -1270,16 +1122,14 @@ $$
 \rm{E}(|X(T) -Y(T)|) \leq C \cdot h^p,
 $$
 for each $h \in [0, \delta]$.&lt;/p&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
+&lt;p&gt;The &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
 for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms.
 For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI)
 can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where &lt;em&gt;b&lt;/em&gt; has only entries on its diagonal or $m=1$.
 The main difficulty for the construction of strong methods with an order &amp;gt; 1/2 arises from the need of an accurate estimation of
 multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of &lt;em&gt;dW&lt;/em&gt; in the case
 of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed
-in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also 
-&lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
+in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also &lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;weak-convergence&#34;&gt;Weak convergence&lt;/h2&gt;
 &lt;p&gt;Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt; in many situations. In these cases, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient and &amp;ndash; due to the less restrictive formulation of the objective &amp;ndash; those solvers are computationally cheaper than their strong counterparts.
 For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics.
@@ -1294,19 +1144,17 @@ $~$&lt;/p&gt;
 &lt;p&gt;for any polynomial $g$ for each $h \in [0, \delta]$.&lt;/p&gt;
 &lt;p&gt;We demonstrate below that &lt;strong&gt;high weak order solvers&lt;/strong&gt; are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.&lt;/p&gt;
 &lt;h2 id=&#34;new-high-weak-order-methods&#34;&gt;New high weak order methods&lt;/h2&gt;
-&lt;p&gt;A list of all new weak solvers is available in the 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
+&lt;p&gt;A list of all new weak solvers is available in the &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
 Note that we also implemented methods designed for the Stratonovich sense.
 For the subsequent examples regarding Ito SDEs, we use only a subset of the plethora of second-order weak solvers.
-We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
-We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
+We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
+We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref2:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref3:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
 The latter is the simplest weak solver, where the Gaussian increments of the strong Euler-Maruyama scheme are replaced by
 two-point distributed random variables with similar moment properties.&lt;/p&gt;
 &lt;p&gt;Rößler&amp;rsquo;s SRK schemes are particularly designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;, since only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn and since the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.
 &lt;code&gt;PL1WM()&lt;/code&gt; in contrast needs to simulate &lt;code&gt;m(m+1)/2&lt;/code&gt; random variables but a smaller number of order conditions needs to be fulfilled.&lt;/p&gt;
 &lt;h3 id=&#34;convergence-tests&#34;&gt;Convergence tests&lt;/h3&gt;
-&lt;p&gt;As in the 
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As in the &lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -1317,64 +1165,61 @@ $$ ~$$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;for the function $f(x)=(x_1)^2$, which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e6&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-using Test
-using Random
-using Plots
-using DiffEqDevTools
-
-function prob_func(prob, i, repeat)
-    remake(prob,seed=seeds[i])
-end
-
-u₀ = [1.0,1.0]
-function f1!(du,u,p,t)
-  @inbounds begin
-    du[1] = -273//512*u[1]
-    du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-  end
-  return nothing
-end
-function g1!(du,u,p,t)
-  @inbounds begin
-    du[1,1] = 1//4*u[1]
-    du[1,2] = 1//16*u[1]
-    du[2,1] = (1-2*sqrt(2))/4*u[1]
-    du[2,2] = 1//10*u[1]+1//16*u[2]
-  end
-  return nothing
-end
-dts = 1 .//2 .^(3: -1:0)
-tspan = (0.0,3.0)
-
-h2(z) = z^2 # but apply it only to u[1]
-
-prob = SDEProblem(f1!,g1!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-numtraj = Int(1e6)
-seed = 100
-Random.seed!(seed)
-seeds = rand(UInt, numtraj)
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h2(sol[end][1]),false),
-        prob_func = prob_func
-        )
-
-sim = test_convergence(dts,ensemble_prob,DRI1(),
-  save_everystep=false,trajectories=numtraj,save_start=false,adaptive=false,
-  weak_timeseries_errors=false,weak_dense_errors=false,
-  expected_value=exp(-3.0)
-  )
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqDevTools&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# but apply it only to u[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Int&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_convergence&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
 Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function for the other aforementioned solvers, we obtain the convergence plot:&lt;/p&gt;
 
 
 
 
 
-  
-
 
 
 
@@ -1385,18 +1230,13 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/weak_conv.png&#34; &gt;
 
-
-  &lt;img src=&#34;https://frankschae.github.io/img/weak_conv.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;weak_conv.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Note that the &lt;code&gt;SimplifiedEM&lt;/code&gt; and the &lt;code&gt;EM&lt;/code&gt; scheme fall on top of each other.
 &lt;code&gt;DRI1()&lt;/code&gt; achieves the smallest errors for a fixed &lt;code&gt;dt&lt;/code&gt; in this study.&lt;/p&gt;
@@ -1404,36 +1244,28 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 &lt;p&gt;Ultimately, we are not only interested in the general convergence slope of an algorithm but also in its speed.
 We&amp;rsquo;d like to select the fastest method depending on the permitted tolerance.
 This is commonly studied using work-precision diagram.
-Thanks to 
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;reltols = 1.0 ./ 4.0 .^ (1:4)
-abstols = reltols#[0.0 for i in eachindex(reltols)]
-setups = [
-          Dict(:alg=&amp;gt;DRI1(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;PL1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;EM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;SimplifiedEM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI2WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false)
-          ]
-test_dt = 1//10000
-appxsol_setup = Dict(:alg=&amp;gt;EM(), :dt=&amp;gt;test_dt)
-wp = @time WorkPrecisionSet(ensemble_prob,
-                            abstols,reltols,setups,test_dt;
-                            maxiters = 1e7,verbose=false,
-                            save_everystep=false,save_start=false,
-                            appxsol_setup = appxsol_setup,
-                            expected_value=exp(-3.0),
-                            trajectories=numtraj, error_estimate=:weak_final)
-plt = plot(wp;legend=:bottomleft)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
-
-  
-
+Thanks to &lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;./&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;4.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;c&#34;&gt;#[0.0 for i in eachindex(reltols)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;PL1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;SimplifiedEM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI2WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10000&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WorkPrecisionSet&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;maxiters&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;verbose&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;error_estimate&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:weak_final&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
@@ -1444,161 +1276,149 @@ plt = plot(wp;legend=:bottomleft)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;WorkPrecision.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Therefore, &lt;code&gt;DRI1&lt;/code&gt; has the best performance in this non-commutative noise case if the error is supposed to stay below 1e-3.
 For larger permitted errors, the &lt;code&gt;SimplifiedEM&lt;/code&gt; scheme might be a good choice. However, the first order methods
 are outclassed when high precision is more of a concern.
-We plan to perform more in-depth benchmarks in the near future what will be reported on the 
-&lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
+We plan to perform more in-depth benchmarks in the near future what will be reported on the &lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
 &lt;h3 id=&#34;adaptive-step-size-control&#34;&gt;Adaptive step-size control&lt;/h3&gt;
-&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
+&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
 &lt;p&gt;To use adaptive step-size control, it is sufficient to set &lt;code&gt;adaptive=true&lt;/code&gt; (default setting). Optionally, one may also pass absolute and relative tolerances.&lt;/p&gt;
 &lt;p&gt;The following julia code simulates the stochastic version of the Brusselator equations with intitial condition&lt;/p&gt;
 &lt;p&gt;$$ X(t=0)=  \begin{pmatrix} 0.1 \\ 0\end{pmatrix},$$&lt;/p&gt;
 &lt;p&gt;on a time span $\mathbb{I}=[0, 100]$ for adaptive (&lt;code&gt;sol&lt;/code&gt;) and fixed step sizes (&lt;code&gt;sol_na&lt;/code&gt;):&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq, DiffEqNoiseProcess, Random
-using Plots
-using DiffEqGPU
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqGPU&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# fix seeds&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.9f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleSummary&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fillalpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;time t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;X(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
-function prob_func(prob, i, repeat)
-  Random.seed!(seeds[i])
-  W = WienerProcess(0.0,0.0,0.0)
-  remake(prob,noise=W)
-end
 
-function brusselator_f!(du,u,p,t)
-  @inbounds begin
-    du[1] = (p[1]-1)*u[1]+p[1]*u[1]^2+(u[1]+1)^2*u[2]
-    du[2] = -p[1]*u[1]-p[1]*u[1]^2-(u[1]+1)^2*u[2]
-  end
-  nothing
-end
 
-function scalar_noise!(du,u,p,t)
-  @inbounds begin
-    du[1] = p[2]*u[1]*(1+u[1])
-    du[2] = -p[2]*u[1]*(1+u[1])
-   end
-   nothing
-end
 
 
-# fix seeds
-seed = 100
-Random.seed!(seed)
-numtraj= 100
-seeds = rand(UInt, numtraj)
-W = WienerProcess(0.0,0.0,0.0)
 
 
-u0 = [-0.1f0,0.0f0]
-tspan = (0.0f0,100.0f0)
-p = [1.9f0,0.1f0]
 
-prob = SDEProblem(brusselator_f!,scalar_noise!,u0,tspan,p,noise=W)
 
-ensembleprob = EnsembleProblem(prob, prob_func = prob_func)
 
-sol = @time solve(ensembleprob,DRI1(),dt=0.1,EnsembleCPUArray(),trajectories=numtraj)
-sol_na = @time solve(ensembleprob,DRI1(),dt=0.8,adaptive=false,EnsembleCPUArray(),trajectories=numtraj)
 
 
-summ = EnsembleSummary(sol,0.0f0:0.5f0:100f0)
-pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quot;X(t)&amp;quot;, label= [&amp;quot;x₁(t)&amp;quot; &amp;quot;x₂(t)&amp;quot;], legend=true)
 
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
 
 
 
 
-  
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator_many_trajectories.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-
-
-
-
-
-
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
+&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref3:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
 as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DifferentialEquations.EnsembleAnalysis
-meansol = timeseries_steps_mean(sol)
-meansol_na = timeseries_point_mean(sol_na,meansol.t)
-dts = []
-tmp1 = tspan[1]
-for tmp2 in meansol.t
-  global tmp1
-  push!(dts,tmp2-tmp1)
-  tmp1 = tmp2
-end
-#
-
-list_plots = []
-for i in 1:length(meansol.u)
-  l = @layout [a  b]
-  plt1 = plot(meansol[1, 1:i],meansol[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;adaptive&amp;quot;,
-         lw=2,
-         linecolor=1
-         )
-  plot!(meansol_na[1, 1:i],meansol_na[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;fixed step size&amp;quot;,
-         lw=2,
-         linecolor=2
-         )
-  pl2 = scatter(dts[1:i], xlabel = &amp;quot;step&amp;quot;, yaxis= &amp;quot;dtᵢ&amp;quot;, xlim = (0, length(meansol.u)),  ylim = (0.0, 2.3), legend=false)
-  plt = plot(plt1, pl2, layout = l)
-  push!(list_plots, plt)
-end
-
-anim = animate(list_plots,lw=2,every=1)
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleAnalysis&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_steps_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_point_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;global&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;#&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@layout&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;adaptive&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;fixed step size&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scatter&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;step&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;dtᵢ&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)),&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;2.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -1610,53 +1430,42 @@ anim = animate(list_plots,lw=2,every=1)
 
 
 
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gpu-usage&#34;&gt;GPU usage&lt;/h3&gt;
 &lt;p&gt;All necessary tools to accelerate the simulation of (stochastic) differential equations on GPUs within the SciML
-ecosystem are collected in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
+ecosystem are collected in the &lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;Currently, bounds checking and return values are not allowed, i.e., functions must be
 written in the form:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f!(du,u,p,t)
-  @inbounds begin
-    du[1] = ..
-  end
-  nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;..&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
 Note that for some high weak order solvers GPU usage is not recommended as scalar indexing is used.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -1665,21 +1474,14 @@ Note that for some high weak order solvers GPU usage is not recommended as scala
       <link>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</link>
       <pubDate>Sat, 30 May 2020 15:10:33 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</guid>
-      <description>&lt;p&gt;First and foremost, I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
+      <description>&lt;p&gt;First and foremost, I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
 Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both  &lt;strong&gt;new high weak order solvers&lt;/strong&gt; as well as &lt;strong&gt;adjoint sensitivity methods&lt;/strong&gt; for stochastic differential equations (SDEs).
-The project is embedded within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
-Ultimately, the planned contributions will allow researchers to simulate (or even 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
+The project is embedded within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
+Ultimately, the planned contributions will allow researchers to simulate (or even &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
 Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.&lt;/p&gt;
 &lt;p&gt;This post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.&lt;/p&gt;
 &lt;h2 id=&#34;high-weak-order-solvers&#34;&gt;High Weak Order Solvers&lt;/h2&gt;
-&lt;p&gt;Currently, the 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
+&lt;p&gt;Currently, the &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
 In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.&lt;/p&gt;
 &lt;p&gt;However in many situations, we are only aiming for computing an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt;.
 In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods.
@@ -1687,7 +1489,7 @@ In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&
 As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.&lt;/p&gt;
 &lt;h3 id=&#34;second-order-runge-kutta-methods-for-ito-sdes&#34;&gt;Second order Runge-Kutta methods for Ito SDEs&lt;/h3&gt;
 &lt;p&gt;In the beginning of the community bonding period I finished the implementations of the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and &lt;code&gt;RI1()&lt;/code&gt;&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;. Specifically, only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn (in contrast to &lt;code&gt;m(m+1)/2&lt;/code&gt; from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.&lt;/p&gt;
-&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -1696,47 +1498,43 @@ $$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e7&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;. The code for a single  &lt;code&gt;dt&lt;/code&gt; reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-numtraj = 1e7
-u₀ = [1.0,1.0]
-function f!(du,u,p,t)
-  du[1] = -273//512*u[1]
-  du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-end
-function g!(du,u,p,t)
-  du[1,1] = 1//4*u[1]
-  du[1,2] = 1//16*u[1]
-  du[2,1] = (1-2*sqrt(2))/4*u[1]
-  du[2,2] = 1//10*u[1]+1//16*u[2]
-end
-dt = 1//8
-tspan = (0.0,10.0)
-prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-h(z) = z^2
-
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h(sol[end][1]),false)
-        )
-sol = solve(ensemble_prob, DRI1();
-        dt=dt,
-        save_start=false,
-        save_everystep=false,
-        weak_timeseries_errors=false,
-        weak_dense_errors=false,
-        trajectories=numtraj)
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
-&lt;pre&gt;&lt;code&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;();&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -1748,29 +1546,21 @@ sol = solve(ensemble_prob, DRI1();
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/DRI1.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/DRI1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;DRI1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In the next couple of weeks, my focus will be on&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;adding other high weak order solvers,&lt;/li&gt;
 &lt;li&gt;implementing adaptive time stepping.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;More of our near-term goals are collected in this 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;More of our near-term goals are collected in this &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;adjoint-sensitivity-methods-for-sdes&#34;&gt;Adjoint Sensitivity Methods for SDEs&lt;/h2&gt;
-&lt;p&gt;In 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
+&lt;p&gt;In &lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
 &lt;p&gt;$$
 L = |f(p)-y|^2.
 $$&lt;/p&gt;
@@ -1779,46 +1569,45 @@ One possibility to obtain the gradient information for (stochastic) differential
 While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.&lt;/p&gt;
 &lt;p&gt;Alternatively to the &amp;ldquo;direct&amp;rdquo; AD approaches, the &lt;strong&gt;adjoint sensitivity method&lt;/strong&gt; can be used&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.
 Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).&lt;/p&gt;
-&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
+&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
 &lt;h3 id=&#34;reverse-sde-time-evolution&#34;&gt;Reverse SDE time evolution&lt;/h3&gt;
 &lt;p&gt;The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms.
 As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed.
 In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values.
-After some fixes for the 
-&lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
+After some fixes for the &lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
 &lt;p&gt;As an example, the code below computes first the forward evolution of an SDE&lt;/p&gt;
 &lt;p&gt;$$ dX  =  \alpha X dt + \beta X dW$$&lt;/p&gt;
 &lt;p&gt;with $\alpha=1.01$, $\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;  using StochasticDiffEq, DiffEqNoiseProcess
-
-  α=1.01
-  β=0.87
-
-  dt = 1e-3
-  tspan = (0.0,1.0)
-  u₀=1/2
-
-  tarray =  collect(tspan[1]:dt:tspan[2])
-
-  f!(du,u,p,t) = du .= α*u
-  g!(du,u,p,t) = du .= β*u
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# to make sure the plot is correct&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NoiseGrid&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-  prob = SDEProblem(f!,g!,[u₀],tspan)
-  sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false)
 
-  _sol = deepcopy(sol) # to make sure the plot is correct
-  W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W))
-  prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1)
-  sol1 = solve(prob1,EulerHeun(),dt=dt)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -1829,19 +1618,10 @@ After some fixes for the
 
 
 
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/animation.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/animation.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;animation.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gradients-of-diagonal-sdes&#34;&gt;Gradients of diagonal SDEs&lt;/h3&gt;
 &lt;p&gt;I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.&lt;/p&gt;
@@ -1855,82 +1635,80 @@ $$&lt;/p&gt;
 &lt;p&gt;acting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\&lt;br&gt;
+\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\
 \frac{d L}{d \beta}  &amp;amp;= 2 \sum_i W(t_i) |X(t_i)|^2
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Test, LinearAlgebra
-using DiffEqSensitivity, StochasticDiffEq
-using Random
-
-seed = 100
-Random.seed!(seed)
-
-u₀ = [0.5]
-tstart = 0.0
-tend = 0.1
-dt = 0.005
-trange = (tstart, tend)
-t = tstart:dt:tend
-tarray = collect(t)
-
-function g(u,p,t)
-  sum(u.^2.0)
-end
-
-function dg!(out,u,p,t,i)
-  (out.=-2.0*u)
-end
-
-p2 = [1.01,0.87]
-
-f(u,p,t) = p[1]*u
-σ(u,p,t) = p[2]*u
-
-
-Random.seed!(seed)
-prob = SDEProblem(f,σ,u₀,trange,p2)
-sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true)
-res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint())
-
-
-noise = vec((@. sol.W(tarray)))
-Wextracted = [W[1][1] for W in noise]
-resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp = [resp1, resp2]
-
-@test isapprox(res_p&#39;, resp, rtol = 1e-6)
-# True
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.005&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:tend&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.=-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RKMil&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;interpretation&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:Stratonovich&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res_u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adjoint_sensitivities&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vec&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;isapprox&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# True&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;finish the current backsolve adjoint version,&lt;/li&gt;
 &lt;li&gt;allow for computing the gradients of non-commuting SDEs,&lt;/li&gt;
 &lt;li&gt;implement also an interpolation adjoint version,&lt;/li&gt;
 &lt;li&gt;benchmark it with respect to AD approaches&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
diff --git a/tag/julia/page/1/index.html b/tag/julia/page/1/index.html
index fd51fc8..970ec39 100644
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diff --git a/tag/machine-learning/index.html b/tag/machine-learning/index.html
index d1d1168..f36ae5d 100644
--- a/tag/machine-learning/index.html
+++ b/tag/machine-learning/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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+  <h1>Machine Learning</h1>
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+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a>
+    </div>
+
+    
+    <a href="/project/ml_for_pt/"  class="summary-link">
+      <div class="article-style">
+        Data-driven methods based on sample instances of the state of a physical system as a function of the system&rsquo;s parameters.
+      </div>
+    </a>
+    
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+    <div class="stream-meta article-metadata">
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+  <span >
+      FS in Collaboration With Julian Arnold</span>, <span >
+      Flemming Holtorf</span>, <span >
+      Eliska Greplova</span>, <span >
+      Agnes Valenti</span>, <span >
+      Martin Zonda</span>, <span >
+      Axel Lode</span>, <span >
+      Gregor Boschung</span>, <span >
+      Sebastian Huber</span>, <span >
+      And Niels Lörch</span>
+      </div>
+      
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.12.031044" target="_blank" rel="noopener">
+    PRX</a>
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+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2311.09128" target="_blank" rel="noopener">
+    NeurIPS Workshop</a>
+
+
+    </div>
+    
+
+  </div>
+  <div class="ml-3">
+    
+    
+      
+      
+      <a href="/project/ml_for_pt/" >
+        <img src="/project/ml_for_pt/featured_hu3194460ba82e7df78553a9c87bbc0c72_237834_311c97c5d06b595a95a2178d807360ed.webp" height="150" width="150"
+            alt="Machine Learning for Phase Transitions" loading="lazy">
+      </a>
+    
+  </div>
+</div>
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+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/publication/pre99/" >F. Schäfer and N. Lörch, Phys. Rev. E 99, 062107 (2019)</a>
+    </div>
+
+    
+    <a href="/publication/pre99/"  class="summary-link">
+      <div class="article-style">
+        We introduce an alternative method to identify phase boundaries in physical systems. It is based on training a predictive model such as …
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
+  PDF
+</a>
+
+
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/frankschae/Vector-field-divergence-of-predictive-model-output-as-indication-of-phase-transitions" target="_blank" rel="noopener">
+  Code
+</a>
+
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@@ -533,120 +1071,111 @@ <h1>Search</h1>
 
 
   
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-      <h2><a href="/publication/pre99/" >F. Schäfer and N. Lörch, Phys. Rev. E 99, 062107 (2019)</a></h2>
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-          We introduce an alternative method to identify phase boundaries in physical systems. It is based on training a predictive model such as a neural network to infer a physical system's parameters from its state. The deviation of the inferred parameters …
-        
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-    </div>
   
 
   
 
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+  
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+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
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diff --git a/tag/many-body-physics/index.html b/tag/many-body-physics/index.html
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-  <h1>Many-body physics</h1>
+  <h1>Many-Body Physics</h1>
 
   
 
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 <div class="universal-wrapper">
   
 
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-      <h2><a href="/project/mbdynamics/" >Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double Well </a></h2>
-      <div class="article-style">
-        
-          Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
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+        Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
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+      FS in Collaboration With Laurent De Forges De Parny</span>, <span >
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diff --git a/tag/many-body-physics/index.xml b/tag/many-body-physics/index.xml
index 1fae598..1afe0d6 100644
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+      <title>Many-Body Physics</title>
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index 91d479d..d98e009 100644
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+<!DOCTYPE html>
+<html lang="en-us">
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+    <title>https://frankschae.github.io/tag/many-body-physics/</title>
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diff --git a/tag/mctdh-x/index.html b/tag/mctdh-x/index.html
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+      
 
       
       
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+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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-  
 
 
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         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
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+      
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+      
+
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     <section class="section-search-results">
 
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+          
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+            
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+            <a class="nav-link " href="/#about"><span>Home</span></a>
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+
+          
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-    
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-      <h2><a href="/publication/entropy/" >F. Schäfer, M. A. Bastarrachea-Magnani, A. U. J. Lode, L. de Forges de Parny, and A. Buchleitner, Entropy 22, 382 (2020)</a></h2>
-      <div class="article-style">
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-          We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact …
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+
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+
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+    </div>
 
     
+    <a href="/project/mbdynamics/"  class="summary-link">
+      <div class="article-style">
+        Study of the spectral structure and the resulting dynamics of a few bosons under consideration of different initial conditions.
+      </div>
+    </a>
     
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+      FS in Collaboration With Laurent De Forges De Parny</span>, <span >
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+      Axel Lode</span>, <span >
+      And Andreas Buchleitner</span>
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+    </div>
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+    <a href="/publication/entropy/"  class="summary-link">
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.mdpi.com/1099-4300/22/4/382" target="_blank" rel="noopener">
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-    © 2020 Frank Schäfer &middot; 
+  
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+  
 
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+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
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-        <pre><code class="tex hljs"></code></pre>
+        
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diff --git a/tag/mctdh-x/index.xml b/tag/mctdh-x/index.xml
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diff --git a/tag/multiple-scattering-theory/index.html b/tag/multiple-scattering-theory/index.html
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-      <h2><a href="/project/scattering/" >Cooperative Scattering of Scalar Waves by Optimized Configurations of Point Scatterers</a></h2>
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-          Numerical optimization of the positions of point scatterers to maximize the total scattering cross section for an incoming plane wave.
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+        Numerical optimization of the positions of point scatterers to maximize the total scattering cross section for an incoming plane wave.
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@@ -686,32 +1038,54 @@ <h2><a href="/project/scattering/" >Cooperative Scattering of Scalar Waves by Op
 
   
   
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       </div>
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index cf03dbe..2bf54e8 100644
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+      <title>Multiple Scattering Theory</title>
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-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
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-          We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering …
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-  <h1>neural networks</h1>
+  <h1>Neural Networks</h1>
 
   
 
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 <div class="universal-wrapper">
   
 
+
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-      <h2><a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a></h2>
-      <div class="article-style">
-        
-          Data-driven methods based on sample instances of the state of a physical system as a function of the system's parameters.
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+      <a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a>
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+    <a href="/project/ml_for_pt/"  class="summary-link">
+      <div class="article-style">
+        Data-driven methods based on sample instances of the state of a physical system as a function of the system&rsquo;s parameters.
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+  <span >
+      FS in Collaboration With Julian Arnold</span>, <span >
+      Flemming Holtorf</span>, <span >
+      Eliska Greplova</span>, <span >
+      Agnes Valenti</span>, <span >
+      Martin Zonda</span>, <span >
+      Axel Lode</span>, <span >
+      Gregor Boschung</span>, <span >
+      Sebastian Huber</span>, <span >
+      And Niels Lörch</span>
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.062107" target="_blank" rel="noopener">
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diff --git a/tag/node/index.xml b/tag/node/index.xml
index a9a49ec..ff9fb86 100644
--- a/tag/node/index.xml
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+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Wed, 10 Mar 2021 00:20:29 +0100</lastBuildDate>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
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diff --git a/tag/node/page/1/index.html b/tag/node/page/1/index.html
index 6ded9d7..22cb7ea 100644
--- a/tag/node/page/1/index.html
+++ b/tag/node/page/1/index.html
@@ -1 +1,10 @@
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\ No newline at end of file
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/tag/node/</title>
+    <link rel="canonical" href="https://frankschae.github.io/tag/node/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/node/">
+  </head>
+</html>
diff --git a/tag/nsde/index.html b/tag/nsde/index.html
index cb3e1e9..c7ec117 100644
--- a/tag/nsde/index.html
+++ b/tag/nsde/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
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-      
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+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
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-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
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+    <link rel="alternate" href="/tag/nsde/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2021-03-10T00:20:29&#43;01:00">
+
   
 
+
   
+  <title>NSDE | FS</title>
 
+  
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+  
 
 
 
-  
 
 
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-  <title>NSDE | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
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   <div class="container">
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         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
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+  <div class="page-header header--fixed">
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+  
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@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
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-          <a class="nav-link " href="/#about"><span>Home</span></a>
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+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
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+                
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+            
+          
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+          
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+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
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+          
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+
+          
+        </ul>
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+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
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@@ -461,65 +681,51 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
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-          
-          
-          
-            
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-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
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+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
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+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
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         </li>
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-
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   </div>
-</nav>
 
+  <div class="page-body">
+    
+    
+    
+
+    
 
-  
 
 
 
@@ -552,89 +758,309 @@ <h1>NSDE</h1>
 <div class="universal-wrapper">
   
 
+
+  
+
   
   
     
-    
-    
-    <div>
-      <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a></h2>
-      <div class="article-style">
-        
-          Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
-        
-      </div>
-    </div>
+
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+
   
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a>
+    </div>
 
     
+    <a href="/project/dp_for_control/"  class="summary-link">
+      <div class="article-style">
+        Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
+      </div>
+    </a>
     
-    
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script>
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+    <div class="stream-meta article-metadata">
+
       
 
       
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-        <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/highlight.min.js" integrity="sha256-eOgo0OtLL4cdq7RdwRUiGKLX9XsIJ7nGhWEKbohmVAQ=" crossorigin="anonymous"></script>
-        
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/languages/julia.min.js"></script>
-        
+
+  <span >
+      FS in Collaboration With Pavel Sekatski</span>, <span >
+      Martin Koppenhöfer</span>, <span >
+      Niels Lörch</span>, <span >
+      Christoph Bruder</span>, <span >
+      And Michal Kloc</span>
+      </div>
       
+    </div>
 
     
+    <div class="btn-links">
+      
+
+
+
+
+
+
+
 
+  
+
+
+
+
+
+
+  
+  
     
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/frankschae/Control-of-Stochastic-Quantum-Dynamics-with-Differentiable-Programming" target="_blank" rel="noopener">
+  Code
+</a>
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
     
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.js" integrity="sha256-EErZamuLefUnbMBQbsEqu1USa+btR2oIlCpBJbyD4/g=" crossorigin="anonymous"></script>
+  
+  
+  
+  
+  
     
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
+    SDE control paper</a>
 
+  
+  
+  
     
+  
+  
+  
+  
+  
     
-    <script>const code_highlighting = true;</script>
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
+    ODE control paper</a>
+
+  
+  
+  
     
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=4" target="_blank" rel="noopener">
+    SIAM CSE 2021 talk</a>
 
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+  
+  
     
+  
+  
+  
+  
+  
     
-    <script>const isSiteThemeDark = false;</script>
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=v8mJVGVdkNQ&amp;list=PLWIVj90xdDE-2eeyFuiooxWcF8kw323Iv&amp;index=3&amp;t=0s" target="_blank" rel="noopener">
+    CMD2020GEFES talk</a>
+
+
+    </div>
     
 
+  </div>
+  <div class="ml-3">
     
     
+      
+      
+      <a href="/project/dp_for_control/" >
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+  
+
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+  
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+  
+  
     
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+  
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+  
 
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+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
+  
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+  <p class="powered-by footer-license-icons">
+    <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank" aria-label="Creative Commons">
+      <i class="fab fa-creative-commons fa-2x" aria-hidden="true"></i>
+      <i class="fab fa-creative-commons-by fa-2x" aria-hidden="true"></i>
+      
+        <i class="fab fa-creative-commons-nc fa-2x" aria-hidden="true"></i>
+      
+      
+        <i class="fab fa-creative-commons-nd fa-2x" aria-hidden="true"></i>
+      
+    </a>
+  </p>
+
+
+
+
+
+  <p class="powered-by">
     
     
-
     
+      
+      
+      
+      
+      
+      
+      Published with <a href="https://hugoblox.com/?utm_campaign=poweredby" target="_blank" rel="noopener">Hugo Blox Builder</a> — the free, <a href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">open source</a> website builder that empowers creators.
     
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+  </p>
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+
+    </div>
+    
+  </div>
+
+  
+
+
+<script src="/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js"></script>
+
+
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+  
+
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+
+
+
+
+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
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+  <script id="search-hit-fuse-template" type="text/x-template">
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@@ -642,42 +1068,18 @@ <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics
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+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
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+  
 
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-    
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@@ -686,32 +1088,54 @@ <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
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+
+
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+
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+<script id="page-data" type="application/json">{"use_headroom":true}</script>
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+  
   
 
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+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
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+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
+
   
 <div id="modal" class="modal fade" role="dialog">
   <div class="modal-dialog">
@@ -723,7 +1147,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -738,5 +1163,25 @@ <h5 class="modal-title">Cite</h5>
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+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
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+
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diff --git a/tag/nsde/index.xml b/tag/nsde/index.xml
index 33f800b..a2ef337 100644
--- a/tag/nsde/index.xml
+++ b/tag/nsde/index.xml
@@ -5,9 +5,9 @@
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+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Wed, 10 Mar 2021 00:20:29 +0100</lastBuildDate>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
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diff --git a/tag/nsde/page/1/index.html b/tag/nsde/page/1/index.html
index 2693d97..63089aa 100644
--- a/tag/nsde/page/1/index.html
+++ b/tag/nsde/page/1/index.html
@@ -1 +1,10 @@
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\ No newline at end of file
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/tag/nsde/</title>
+    <link rel="canonical" href="https://frankschae.github.io/tag/nsde/">
+    <meta name="robots" content="noindex">
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+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/nsde/">
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+</html>
diff --git a/tag/numerical-optimization/index.html b/tag/numerical-optimization/index.html
index 7ce6da8..7381798 100644
--- a/tag/numerical-optimization/index.html
+++ b/tag/numerical-optimization/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
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-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
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+<head>
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+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
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-    
-    
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-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
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-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tag/numerical-optimization/">
+
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+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
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-  <meta name="theme-color" content="#2962ff">
+    
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+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
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-          We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering …
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+  
   
 
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         </button>
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       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
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+        <pre><code></code></pre>
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         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
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diff --git a/tag/numerical-optimization/index.xml b/tag/numerical-optimization/index.xml
index d1248bd..338f6bb 100644
--- a/tag/numerical-optimization/index.xml
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+    <description>Numerical Optimization</description>
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-      <title>numerical optimization</title>
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+      <title>Numerical Optimization</title>
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+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/tag/numerical-optimization/</title>
+    <link rel="canonical" href="https://frankschae.github.io/tag/numerical-optimization/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/numerical-optimization/">
+  </head>
+</html>
diff --git a/tag/phase-transitions/index.html b/tag/phase-transitions/index.html
index ce0488b..3d56598 100644
--- a/tag/phase-transitions/index.html
+++ b/tag/phase-transitions/index.html
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-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
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-  
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+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +141,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
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-
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-      
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+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
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-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
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+
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+
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+
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+  <title>Phase Transitions | FS</title>
 
+  
+  
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-  
 
 
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         </div>
         <div class="col-6 col-search-close">
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         </div>
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       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
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+  <div class="page-header header--fixed">
+  
+  
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+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
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+        <ul class="navbar-nav d-md-inline-flex">
           
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+            
           
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-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
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-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
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+          
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+
+          
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-          
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diff --git a/tag/quantum-control/index.html b/tag/quantum-control/index.html
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+  <title>Quantum Control | FS</title>
 
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@@ -298,6 +430,12 @@ <h1>Search</h1>
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+  
+
+  
+  
+  
+    
+  
+<div class="universal-wrapper pt-3">
+  <h1>Quantum Control</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  
+
+
+  
+
+  
+  
+    
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+  
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a>
+    </div>
+
+    
+    <a href="/project/controlbounds/"  class="summary-link">
+      <div class="article-style">
+        We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+      <div>
         
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
-        </li>
+  <span >
+      FS in Collaboration With Flemming Holtorf</span>, <span >
+      Julian Arnold</span>, <span >
+      Chris Rackauckas</span>, <span >
+      And Alan Edelman</span>
+      </div>
+      
+    </div>
+
+    
+    <div class="btn-links">
+      
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+  
+  
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+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2304.03366" target="_blank" rel="noopener">
+    arXiv preprint</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://ieeexplore.ieee.org/abstract/document/10313626" target="_blank" rel="noopener">
+    QCE</a>
+
+
+    </div>
+    
+
+  </div>
+  <div class="ml-3">
+    
+    
+      
+      
+      <a href="/project/controlbounds/" >
+        <img src="/project/controlbounds/featured_hu9a224a6f0b76f74a72bd3d29668f06be_15378_efa414e40070ef54cd7713cdec8327e3.webp" height="91" width="150"
+            alt="Performance Bounds for Quantum Control" loading="lazy">
+      </a>
+    
+  </div>
+</div>
+
+  
+    
+
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+  
+
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a>
+    </div>
+
+    
+    <a href="/project/dp_for_control/"  class="summary-link">
+      <div class="article-style">
+        Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+      <div>
+        
+
+  <span >
+      FS in Collaboration With Pavel Sekatski</span>, <span >
+      Martin Koppenhöfer</span>, <span >
+      Niels Lörch</span>, <span >
+      Christoph Bruder</span>, <span >
+      And Michal Kloc</span>
+      </div>
+      
+    </div>
+
+    
+    <div class="btn-links">
+      
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+  
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+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/frankschae/Control-of-Stochastic-Quantum-Dynamics-with-Differentiable-Programming" target="_blank" rel="noopener">
+  Code
+</a>
+
+
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+
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+
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+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
+    SDE control paper</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
+    ODE control paper</a>
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=4" target="_blank" rel="noopener">
+    SIAM CSE 2021 talk</a>
 
-        
-        
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=v8mJVGVdkNQ&amp;list=PLWIVj90xdDE-2eeyFuiooxWcF8kw323Iv&amp;index=3&amp;t=0s" target="_blank" rel="noopener">
+    CMD2020GEFES talk</a>
 
-      
 
-        
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     </div>
+    
 
-    <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+  </div>
+  <div class="ml-3">
+    
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-        <a class="nav-link js-search" href="#"><i class="fas fa-search" aria-hidden="true"></i></a>
-      </li>
       
+      <a href="/project/dp_for_control/" >
+        <img src="/project/dp_for_control/featured_hudf85e1c26ad74a07a00aa9287e088b01_236898_6e426bbd03dcc15fb72201019eec9119.webp" height="113" width="150"
+            alt="Control of (Stochastic) Quantum Dynamics with Differentiable Programming" loading="lazy">
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+    
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+  <div class="page-footer">
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@@ -533,120 +1084,111 @@ <h1>Search</h1>
 
 
   
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-  <h1>quantum control</h1>
 
-  
 
-  
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-<div class="universal-wrapper">
+
+
+
+
+
+
+
+
+  
+  
   
 
   
   
     
-    
-    
-    <div>
-      <h2><a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a></h2>
-      <div class="article-style">
-        
-          We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
-        
-      </div>
-    </div>
+  
   
     
-    
-    
-    <div>
-      <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a></h2>
-      <div class="article-style">
-        
-          Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
-        
-      </div>
-    </div>
   
 
   
 
-</div>
+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
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+  </p>
+</footer>
 
+    </div>
     
-    
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.js" integrity="sha256-EErZamuLefUnbMBQbsEqu1USa+btR2oIlCpBJbyD4/g=" crossorigin="anonymous"></script>
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+<script src="/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js"></script>
 
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+  
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+  
+
+
+
+
+
+
+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
+
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+  
+  <script id="search-hit-fuse-template" type="text/x-template">
+    <div class="search-hit" id="summary-{{key}}">
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@@ -654,42 +1196,18 @@ <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics
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+  
 
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-    
 
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-      
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-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
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@@ -698,32 +1216,54 @@ <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
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-    
-    <span class="float-right" aria-hidden="true">
-      <a href="#" class="back-to-top">
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+
+
+
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+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
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+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
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+  
   
 
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+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
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@@ -735,7 +1275,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -750,5 +1291,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
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 </body>
 </html>
diff --git a/tag/quantum-control/index.xml b/tag/quantum-control/index.xml
index f25358e..591ce92 100644
--- a/tag/quantum-control/index.xml
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 <rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
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+    <title>Quantum Control | FS</title>
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+    <description>Quantum Control</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sat, 29 Apr 2023 17:26:49 -0400</lastBuildDate>
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-      <title>quantum control</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Quantum Control</title>
       <link>https://frankschae.github.io/tag/quantum-control/</link>
     </image>
     
diff --git a/tag/quantum-control/page/1/index.html b/tag/quantum-control/page/1/index.html
index e48902d..0b5af6c 100644
--- a/tag/quantum-control/page/1/index.html
+++ b/tag/quantum-control/page/1/index.html
@@ -1 +1,10 @@
-<!DOCTYPE html><html><head><title>https://frankschae.github.io/tag/quantum-control/</title><link rel="canonical" href="https://frankschae.github.io/tag/quantum-control/"/><meta name="robots" content="noindex"><meta charset="utf-8" /><meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/quantum-control/" /></head></html>
\ No newline at end of file
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/tag/quantum-control/</title>
+    <link rel="canonical" href="https://frankschae.github.io/tag/quantum-control/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/quantum-control/">
+  </head>
+</html>
diff --git a/tag/sciml/index.html b/tag/sciml/index.html
index 0b2a4cd..23bbca2 100644
--- a/tag/sciml/index.html
+++ b/tag/sciml/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tag/sciml/">
+
+
+
+
+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
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+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
+
   
   
   
     
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+    
+
     
     
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+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
+  
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+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
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+  
+  
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+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
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+  
 
   
+  
+
+
+
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+
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+
+
+
+
+
+
 
 
 
 
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-
   
-  <link rel="alternate" href="/tag/sciml/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
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+<link rel="canonical" href="https://frankschae.github.io/tag/sciml/" />
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+  <link rel="manifest" href="/manifest.webmanifest" />
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+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
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+
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+  <meta property="twitter:creator" content="@GetResearchDev" />
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+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/tag/sciml/" />
+<meta property="og:title" content="SciML | FS" />
+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
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+    <meta property="og:updated_time" content="2021-03-10T00:20:29&#43;01:00" />
   
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-    
-    
   
+
   
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-    <meta property="og:updated_time" content="2021-03-10T00:20:29&#43;01:00">
+
   
 
+
   
+  <title>SciML | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>SciML | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
+
+  
+  
+  
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   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
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         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
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       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
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     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
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@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
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-  <div class="container">
 
-    
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-      <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a></h2>
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-          Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
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+      <a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics with Differentiable Programming</a>
+    </div>
 
     
+    <a href="/project/dp_for_control/"  class="summary-link">
+      <div class="article-style">
+        Quantum control based on parametrized controllers trained with gradient information computed by (adjoint) sensitivity methods.
+      </div>
+    </a>
     
-    
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+  <span >
+      FS in Collaboration With Pavel Sekatski</span>, <span >
+      Martin Koppenhöfer</span>, <span >
+      Niels Lörch</span>, <span >
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+      And Michal Kloc</span>
+      </div>
       
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+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/frankschae/Control-of-Stochastic-Quantum-Dynamics-with-Differentiable-Programming" target="_blank" rel="noopener">
+  Code
+</a>
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-      <script src="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.js" integrity="sha256-EErZamuLefUnbMBQbsEqu1USa+btR2oIlCpBJbyD4/g=" crossorigin="anonymous"></script>
+  
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+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://arxiv.org/abs/2101.01190" target="_blank" rel="noopener">
+    SDE control paper</a>
 
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+  
     
+  
+  
+  
+  
+  
     
-    <script>const code_highlighting = true;</script>
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://iopscience.iop.org/article/10.1088/2632-2153/ab9802" target="_blank" rel="noopener">
+    ODE control paper</a>
+
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+  
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+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=uDUwdAqKzYM&amp;list=PLP8iPy9hna6TxktMt-IzdU2vQpGp3bwDn&amp;index=4" target="_blank" rel="noopener">
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+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="https://www.youtube.com/watch?v=v8mJVGVdkNQ&amp;list=PLWIVj90xdDE-2eeyFuiooxWcF8kw323Iv&amp;index=3&amp;t=0s" target="_blank" rel="noopener">
+    CMD2020GEFES talk</a>
+
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+    </div>
     
 
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+  <div class="ml-3">
     
     
+      
+      
+      <a href="/project/dp_for_control/" >
+        <img src="/project/dp_for_control/featured_hudf85e1c26ad74a07a00aa9287e088b01_236898_6e426bbd03dcc15fb72201019eec9119.webp" height="113" width="150"
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+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
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+      Published with <a href="https://hugoblox.com/?utm_campaign=poweredby" target="_blank" rel="noopener">Hugo Blox Builder</a> — the free, <a href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">open source</a> website builder that empowers creators.
     
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@@ -686,32 +1088,54 @@ <h2><a href="/project/dp_for_control/" >Control of (Stochastic) Quantum Dynamics
 
   
   
-  <div class="container">
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+  
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 <div id="modal" class="modal fade" role="dialog">
   <div class="modal-dialog">
@@ -723,7 +1147,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -738,5 +1163,25 @@ <h5 class="modal-title">Cite</h5>
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diff --git a/tag/sciml/index.xml b/tag/sciml/index.xml
index 3268bf3..b3e62ae 100644
--- a/tag/sciml/index.xml
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+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Wed, 10 Mar 2021 00:20:29 +0100</lastBuildDate>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
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diff --git a/tag/sciml/page/1/index.html b/tag/sciml/page/1/index.html
index 28fbd9c..b2b1e4e 100644
--- a/tag/sciml/page/1/index.html
+++ b/tag/sciml/page/1/index.html
@@ -1 +1,10 @@
-<!DOCTYPE html><html><head><title>https://frankschae.github.io/tag/sciml/</title><link rel="canonical" href="https://frankschae.github.io/tag/sciml/"/><meta name="robots" content="noindex"><meta charset="utf-8" /><meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/sciml/" /></head></html>
\ No newline at end of file
+<!DOCTYPE html>
+<html lang="en-us">
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+    <title>https://frankschae.github.io/tag/sciml/</title>
+    <link rel="canonical" href="https://frankschae.github.io/tag/sciml/">
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+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/sciml/">
+  </head>
+</html>
diff --git a/tag/shadowing/index.html b/tag/shadowing/index.html
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+++ b/tag/shadowing/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
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+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
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-    <meta property="og:updated_time" content="2021-07-02T11:08:22&#43;02:00">
+
   
 
+
   
+  <title>Shadowing | FS</title>
 
+  
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+  
 
 
 
-  
 
 
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-  <title>Shadowing | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
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         <div class="col-6 col-search-close">
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+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
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+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
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     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
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+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
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+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
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+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
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+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
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+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
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@@ -461,65 +681,51 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
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-          
-          
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-            
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-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
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+              <span>Automatic</span>
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+</header>
 
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-</nav>
 
+  <div class="page-body">
+    
+    
+    
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+    
 
-  
 
 
 
@@ -552,89 +758,231 @@ <h1>Shadowing</h1>
 <div class="universal-wrapper">
   
 
+
   
+
   
-    
-    
-    
-    <div>
-      <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a></h2>
-      <div class="article-style">
-        
-          In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
-        
-      </div>
-    </div>
   
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+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems</a>
+    </div>
 
     
+    <a href="/post/shadowing/"  class="summary-link">
+      <div class="article-style">
+        In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters.
+      </div>
+    </a>
     
-    
-      <script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script>
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@@ -642,42 +990,18 @@ <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensit
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@@ -686,32 +1010,54 @@ <h2><a href="/post/shadowing/" >Shadowing Methods for Forward and Adjoint Sensit
 
   
   
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@@ -738,5 +1085,25 @@ <h5 class="modal-title">Cite</h5>
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diff --git a/tag/shadowing/index.xml b/tag/shadowing/index.xml
index e1c08d7..57e8c61 100644
--- a/tag/shadowing/index.xml
+++ b/tag/shadowing/index.xml
@@ -5,9 +5,9 @@
     <link>https://frankschae.github.io/tag/shadowing/</link>
       <atom:link href="https://frankschae.github.io/tag/shadowing/index.xml" rel="self" type="application/rss+xml" />
     <description>Shadowing</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Fri, 02 Jul 2021 11:08:22 +0200</lastBuildDate>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Fri, 02 Jul 2021 11:08:22 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
       <title>Shadowing</title>
       <link>https://frankschae.github.io/tag/shadowing/</link>
     </image>
@@ -18,8 +18,7 @@
       <pubDate>Fri, 02 Jul 2021 11:08:22 +0200</pubDate>
       <guid>https://frankschae.github.io/post/shadowing/</guid>
       <description>&lt;p&gt;In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the &amp;ldquo;butterfly effect&amp;rdquo;. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.&lt;/p&gt;
-&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
+&lt;p&gt;Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; &lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; (LSS) or non-intrusive least squares shadowing&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt; (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem &amp;ndash; the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity.jl&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:&lt;/p&gt;
 &lt;p&gt;$$
 \langle g \rangle_∞ = \lim_{T \rightarrow ∞} \langle g \rangle_T,
@@ -33,57 +32,59 @@ Under the assumption of ergodicity, $\langle g \rangle_∞$ only depends on $p$.
 &lt;p&gt;One of the most important chaotic models is the Lorenz system which is a simplified model for atmospheric convection. The Lorenz system has three states $x$, $y$, and $z$, as well as three parameters $\rho$, $\sigma$, and $\beta$. Its time evolution is given by the ODE:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{pmatrix}
-\text{d}x \\&lt;br&gt;
-\text{d}y \\&lt;br&gt;
-\text{d}z \\&lt;br&gt;
+\text{d}x \\
+\text{d}y \\
+\text{d}z \\
 \end{pmatrix} = \begin{pmatrix}
-\sigma (y-x)\\&lt;br&gt;
-x(\rho-z) - y\\&lt;br&gt;
-x y - \beta z \\&lt;br&gt;
+\sigma (y-x)\\
+x(\rho-z) - y\\
+x y - \beta z \\
 \end{pmatrix}\text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;For simplicity, let us fix $\sigma=10$ and $\beta=8/3$ and focus only on the sensitivity with respect to $\rho$. The classic Lorenz attractor is obtained when using $\rho=28$:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Random; Random.seed!(1234)
-using OrdinaryDiffEq
-using Statistics
-using QuadGK, ForwardDiff, Calculus
-using DiffEqSensitivity
-using SparseArrays, LinearAlgebra
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;OrdinaryDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Statistics&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;QuadGK&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SparseArrays&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# simulate 1 trajectory of the Lorenz system forward&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;28.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;30.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LaTeXStrings&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;initial&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;attractor&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;30&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;49&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_forward.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-# simulate 1 trajectory of the Lorenz system forward
-function lorenz!(du,u,p,t)
-  du[1] = 10*(u[2]-u[1])
-  du[2] = u[1]*(p[1]-u[3]) - u[2]
-  du[3] = u[1]*u[2] - (8//3)*u[3]
-end
 
-p = [28.0]
-tspan_init = (0.0,30.0)
-tspan_attractor = (30.0,50.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
 
-using Plots, LaTeXStrings
-pl1 = plot(sol_init,vars=(1,2,3), legend=true,
-  label = &amp;quot;initial&amp;quot;,
-  labelfontsize=20,
-  lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-plot!(pl1, sol_attractor,vars=(1,2,3), label=&amp;quot;attractor&amp;quot;,xlims=(-25,30),ylims=(-30,30),zlims=(5,49)
- )
-savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
 
 
 
 
 
-  
 
 
 
@@ -93,81 +94,66 @@ savefig(pl1, &amp;quot;Lorenz_forward.png&amp;quot;)
 
 
 
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_forward.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_forward.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Here, we separated the trajectory in two parts: We plot the initial transient dynamics starting from random initial conditions towards the attractor in blue and the subsequent time evolution lying entirely on the attractor in orange.&lt;/p&gt;
-&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
+&lt;p&gt;Following Refs.&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref1:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, we choose&lt;/p&gt;
 &lt;p&gt;$$
 \langle z \rangle_∞ = \lim_{T \rightarrow ∞} \frac{1}{T} \int_0^T z \text{d}t
 $$&lt;/p&gt;
 &lt;p&gt;as the objective, where we only use the trajectory that lies completely on the attractor (i.e., the orange trajectory in the plot on top). Let us first study the objective as a function of $\rho$.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function compute_objective(sol)
-  quadgk(t-&amp;gt; sol(t)[end]/(tspan_attractor[2]-tspan_attractor[1]) ,tspan_attractor[1],tspan_attractor[2], atol=1e-14, rtol=1e-10)[1]
-end
-
-pl2 = plot(sol_attractor.t, getindex.(sol_attractor.u,3), ylabel=L&amp;quot;z(t)&amp;quot;, xlabel=L&amp;quot;t&amp;quot;, label=false, labelfontsize=20,lw = 2)
-mean_z = [mean(getindex.(sol_attractor.u,3))]
-int_z = compute_objective(sol_attractor)
-hline!(pl2, [int_z], label=L&amp;quot;\langle z\rangle&amp;quot;, lw = 2)
-savefig(pl2, &amp;quot;zsingle.png&amp;quot;)
-
-# for each value of the parameter, solve 20 times the initial value problem
-# wrap the procedure inside a function depending on p
-function Lorenz_solve(p)
-  u0 = rand(3)
-  prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-  sol_init = solve(prob_init,Tsit5())
-  prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-  sol_attractor = solve(prob_attractor,Vern9(),abstol=1e-14,reltol=1e-14)
-  sol_attractor, prob_attractor
-end
-
-Niter = 10
-ps = collect(0.0:1.0:50.0)
-probs = []
-sols = []
-zmean = []
-zstd = []
-for ρ in ps
-  @show ρ
-  ztmp = []
-  for i=1:Niter
-    sol, prob = Lorenz_solve([ρ])
-    zbar = compute_objective(sol)
-    push!(sols, sol)
-    push!(probs, prob)
-    push!(ztmp, zbar)
-  end
-  push!(zmean,mean(ztmp))
-  push!(zstd,std(ztmp))
-end
-
-pl3 = plot(ps,zmean, ribbon = zstd, ylabel=L&amp;quot;\langle z\rangle&amp;quot;, xlabel=L&amp;quot;\rho&amp;quot;, legend=false, labelfontsize=20, lw = 2, xlims=(0,50),ylims=(0,50))
-savefig(pl3, &amp;quot;zvsrho.png&amp;quot;)
-
-pl4 = plot(pl2,pl3, margin=3Plots.mm, layout = (1, 2), size=(600,300))
-savefig(pl4, &amp;quot;z.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We obtain:&lt;/p&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;quadgk&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;atol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-10&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;mean_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;hline!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;int_z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zsingle.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# for each value of the parameter, solve 20 times the initial value problem&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# wrap the procedure inside a function depending on p&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Niter&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Lorenz_solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;([&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ρ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;probs&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zbar&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;std&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ztmp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ps&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zmean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ribbon&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zstd&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;\langle z&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;angle&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;&lt;/span&gt;&lt;span class=&#34;se&#34;&gt;\r&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;ho&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;50&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;zvsrho.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;margin&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;mm&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;size&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;600&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;300&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;z.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We obtain:&lt;/p&gt;
 
 
-  
 
 
 
@@ -179,40 +165,35 @@ savefig(pl4, &amp;quot;z.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/z.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/z.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;z.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;That is, we find a slope of approximately one (almost everywhere except at the kink $\rho\approx 23$), and, therefore, we expect a sensitivity of&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}\langle z \rangle_∞}{\text{d} \rho} \approx 1.
 $$&lt;/p&gt;
 &lt;h2 id=&#34;conventional-forward-mode-sensitivity-analysis-and-finite-differencing&#34;&gt;Conventional forward-mode sensitivity analysis and finite-differencing&lt;/h2&gt;
-&lt;p&gt;For non-chaotic systems, we would just use the 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function G(p, prob=prob_attractor)
-  tmp_prob = remake(prob,p=p)
-  tmp_sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
-  res = compute_objective(tmp_sol)
-  @info res
-  res
-end
-sense_forward = ForwardDiff.gradient(G,p)
-sense_calculus = Calculus.gradient(G,p)
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;we find diverging values:&lt;/p&gt;
+&lt;p&gt;For non-chaotic systems, we would just use the &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/sensitivity/#Sensitivity-Algorithms&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;standard discrete or continuous forward sensitivity methods&lt;/a&gt; or even finite-differencing.  If we try to compute the sensitivity for the Lorenz system:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;compute_objective&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp_sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@info&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;res&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_forward&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardDiff&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sense_calculus&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Calculus&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;G&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;we find diverging values:&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\&lt;br&gt;
+&amp;amp; \frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx -49899 {\text{ (ForwardDiff)}}  \\
 &amp;amp;\frac{\text{d}\langle z \rangle_\infty}{\text{d} \rho} \Bigg\rvert_{\rho=28} \approx 472 {\text{ (Calculus)}}
 \end{aligned}
 $$&lt;/p&gt;
@@ -220,44 +201,39 @@ $$&lt;/p&gt;
 &lt;p&gt;$$
 \frac{\text{d}}{\text{d} \rho} \langle z \rangle_∞ \neq \lim_{T \rightarrow ∞} \frac{\partial}{\partial \rho} \langle z \rangle_T
 $$&lt;/p&gt;
-&lt;p&gt;Similarly, using 
-&lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;prob_attractor1 = ODEProblem(lorenz!,sol_init[end],(0.0, 50.0),p)
-prob_attractor2 = ODEProblem(lorenz!,convert.(Float32, sol_init[end]),(0f0, 50f0),convert.(Float32,p))
-sol1 = solve(prob_attractor1,Tsit5(),abstol=1e-6,reltol=1e-6, saveat=0.01)
-sol2 = solve(prob_attractor2,Tsit5(),abstol=1f-6,reltol=1f-6, saveat=0.01f0)
-
-list_plots = []
-t1 = 0.0
-for i in 1:500
-  t2 = i*0.1
-  plt1 = plot(sol1, vars=(1,2,3), tspan=(t1,t2), denseplot=true, legend=true,
-     label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-     xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-     xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  plot!(plt1, sol2,vars=(1,2,3), tspan=(t1,t2), denseplot=true, label=&amp;quot;Float32&amp;quot;,
-        xlims=(-20,25),ylims=(-28,25),zlims=(5,48))
-  push!(list_plots, plt1)
-end
-
-anim = animate(list_plots,every=1)
-
-pl1 = plot(sol1,vars=(1,2,3), legend=true,
-  label = &amp;quot;Float64&amp;quot;, labelfontsize=20, lw = 2,
-  xlabel = L&amp;quot;x&amp;quot;, ylabel = L&amp;quot;y&amp;quot;, zlabel = L&amp;quot;z&amp;quot;,
-  xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-plot!(pl1, sol2,vars=(1,2,3), label=&amp;quot;Float32&amp;quot;, xlims=(-20,25),ylims=(-28,25),zlims=(5,48)
- )
-
-savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;p&gt;Similarly, using &lt;a href=&#34;https://diffeq.sciml.ai/stable/analysis/uncertainty_quantification/#Example-3:-Adaptive-ProbInts-on-the-Lorenz-Attractor&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;uncertainty quantification&lt;/a&gt; one realizes that due to finite numerical precision and the associated unavoidable errors that are amplified exponentially, one cannot follow the true solution of a chaotic system for long times. We can visualize this by solving the Lorenz system twice with exactly the same parameters and initial condition but with different floating point number precision. In the following animation, we see an $O(1)$ difference between both trajectories after a few Lyapunov lengths:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;50f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;convert&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;Float32&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1f-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;500&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;     &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;denseplot&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Float64&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;labelfontsize&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ylabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;y&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;zlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;sa&#34;&gt;L&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;z&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;vars&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;Float32&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ylims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;28&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;25&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zlims&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;48&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt; &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;savefig&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;pl1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;Lorenz_Floats.png&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -269,18 +245,14 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Without animation:&lt;/p&gt;
 
@@ -288,72 +260,61 @@ savefig(pl1, &amp;quot;Lorenz_Floats.png&amp;quot;)
 
 
 
-  
-
-
-
-
-
 
 
 
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Lorenz_Floats.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
 
 
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Lorenz_Floats.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-&lt;p&gt;Luckily, the 
-&lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
+&lt;p&gt;Luckily, the &lt;a href=&#34;https://mathworld.wolfram.com/ShadowingTheorem.html&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;shadowing lemma&lt;/a&gt; states:&lt;/p&gt;
 &lt;blockquote&gt;
 &lt;p&gt;Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near (&amp;ldquo;shadows&amp;rdquo;) the numerically computed one.&lt;/p&gt;
 &lt;/blockquote&gt;
 &lt;h2 id=&#34;shadowing-methods&#34;&gt;Shadowing methods&lt;/h2&gt;
-&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;p&gt;The central idea of the shadowing methods is to distill the long-time effect (which actually shifts the attractor) due to a variation of the system parameters (upwards in the $z$-direction with increasing $\rho$ for the Lorenz system) from the transient effect, i.e., the butterfly effect that looks like exponentially diverging trajectories due to variations of the initial conditions.  That implies that we aim at finding two trajectories, one with $p$ and one with $p+\delta p$, which do &lt;em&gt;not&lt;/em&gt; diverge exponentially from each other (which exist thanks to the shadowing lemma). In this case, their difference will only contain the long-time effect. More details can be found in Refs. &lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; and &lt;sup id=&#34;fnref2:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;, including a visualization of both effects in Fig. 1 of Ref. &lt;sup id=&#34;fnref3:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;lss-and-nilss-for-the-lorenz-system&#34;&gt;LSS and NILSS for the Lorenz system&lt;/h2&gt;
-&lt;p&gt;Switching to LSS or NILSS within the 
-&lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# objective
-g(u,p,t) = u[end]
-
-####
-# LSS
-####
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.CosWindowing()), g)
-@show shadow_forward(lss_problem) # 1.0095888187322035
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=DiffEqSensitivity.Cos2Windowing()), g)
-@show shadow_forward(lss_problem) # 1.0343951385924328
-
-lss_problem = ForwardLSSProblem(sol_attractor, ForwardLSS(alpha=10.0), g)
-@show shadow_forward(lss_problem) # 1.0284286902740765
-
-adjointlss_problem = AdjointLSSProblem(sol_attractor, AdjointLSS(alpha=10.0), g)
-@show shadow_adjoint(adjointlss_problem) # 1.028428690274077
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;# select via sensealg in solve
-using Zygote
-
-function GLSS(p; sensealg=ForwardLSS(), dt=0.01, g=nothing)
-  _prob = remake(prob_attractor,p=p)
-  _sol = solve(_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=dt,sensealg=sensealg, g=g)
-  sum(getindex.(_sol.u,3))
-end
-
-dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
+&lt;p&gt;Switching to LSS or NILSS within the &lt;a href=&#34;https://sciml.ai/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem is straightforward by either defining the associated LSS (&lt;code&gt;ForwardLSSProblem&lt;/code&gt; or &lt;code&gt;AdjointLSSProblem&lt;/code&gt;) or NILSS problem (&lt;code&gt;NILSSProblem&lt;/code&gt;) type manually:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# objective&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# LSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;CosWindowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0095888187322035&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Cos2Windowing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;()),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0343951385924328&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0284286902740765&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;AdjointLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;alpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;shadow_adjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adjointlss_problem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.028428690274077&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;or by setting the &lt;code&gt;sensealg=&lt;/code&gt; kwarg in &lt;code&gt;solve()&lt;/code&gt;:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# select via sensealg in solve&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ForwardLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Vern9&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;abstol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e-14&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;saveat&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;getindex&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Zygote&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;gradient&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;GLSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9694728321500617&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Note that we have implemented three different options for forward shadowing with &lt;code&gt;LSS()&lt;/code&gt;:&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;&lt;code&gt;CosWindowing()&lt;/code&gt; (default)&lt;/li&gt;
 &lt;li&gt;&lt;code&gt;Cos2Windowing()&lt;/code&gt;&lt;/li&gt;
@@ -362,72 +323,68 @@ dp1 = Zygote.gradient((p)-&amp;gt;GLSS(p),p) # 0.9694728321500617
 &lt;p&gt;Additionally, an adjoint implementation &lt;code&gt;AdjointLSS()&lt;/code&gt; is available that is particularly recommended for a large number of system parameters.  Based on the values computed above, we can easily check that &lt;code&gt;AdjointLSS(alpha=10.0)&lt;/code&gt; agrees perfectly with &lt;code&gt;ForwardLSS(alpha=10.0)&lt;/code&gt;. In all cases considered, we find the expected sensitivity value of $\approx 1$.&lt;/p&gt;
 &lt;p&gt;However, the use of &lt;code&gt;LSS()&lt;/code&gt; is (typically) much more expensive than the use of &lt;code&gt;NILSS()&lt;/code&gt;, because &lt;code&gt;LSS()&lt;/code&gt; needs to solve a large linear system. This linear system scales with the number of independent variables in the differential equation times the number of time steps and, thus, it can become very large.  The computational and memory costs of &lt;code&gt;NILSS()&lt;/code&gt; scale with the number of positive (unstable) Lyapunov exponents, since it constrains the optimization problem in the LSS method to its unstable subspace. In many cases, this number is much smaller than the number of independent variables, hence making &lt;code&gt;NILSS()&lt;/code&gt; more efficient.&lt;/p&gt;
 &lt;p&gt;In the &lt;code&gt;NILSS()&lt;/code&gt; algorithm, the user can control the number of steps per segment as well as the number of segments.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;####
-# NILSS
-####
-
-# make sure trajectory is fully on the attractor
-Random.seed!(1234)
-tspan_init = (0.0,100.0)
-tspan_attractor = (100.0,120.0)
-u0 = rand(3)
-prob_init = ODEProblem(lorenz!,u0,tspan_init,p)
-sol_init = solve(prob_init,Tsit5())
-prob_attractor = ODEProblem(lorenz!,sol_init[end],tspan_attractor,p)
-
-nseg = 100 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 0.9966924374966089
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;nseg = 20 # number of segments on time interval
-nstep = 2001 # number of steps on each segment
-
-nilss_prob = NILSSProblem(prob_attractor, NILSS(nseg, nstep), g)
-@show DiffEqSensitivity.shadow_forward(nilss_prob,Tsit5()) # 1.0416028730638789
-
-# Warning: Detected a large value of ξ at the beginning of a segment.
-# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# NILSS&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;####&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# make sure trajectory is fully on the attractor&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1234&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;120.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;ODEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lorenz!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_init&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 0.9966924374966089&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;If the number of segments is chosen too small, a warning is thrown:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;20&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of segments on time interval&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2001&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# number of steps on each segment&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSSProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob_attractor&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NILSS&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nseg&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;nstep&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@show&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;shadow_forward&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;nilss_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Tsit5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# 1.0416028730638789&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# Warning: Detected a large value of ξ at the beginning of a segment.&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# └ @ DiffEqSensitivity ~/.julia/dev/DiffEqSensitivity/src/nilss.jl:474&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;In the future, we might add an option for the automate control of these variables following the proposal in the NILSS paper&lt;sup id=&#34;fnref4:5&#34;&gt;&lt;a href=&#34;#fn:5&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;5&lt;/a&gt;&lt;/sup&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;outlook&#34;&gt;Outlook&lt;/h2&gt;
 &lt;p&gt;With respect to the shadowing methods for chaotic systems, we are planning to implement further methods, such as&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;NILSAS&lt;sup id=&#34;fnref:6&#34;&gt;&lt;a href=&#34;#fn:6&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;6&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;li&gt;FD-NILSS&lt;sup id=&#34;fnref:7&#34;&gt;&lt;a href=&#34;#fn:7&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;7&lt;/a&gt;&lt;/sup&gt;&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For further information and a collection of other methods, the interested reader is invited to track the corresponding &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues/102&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt; in the DiffEqSensitivity.jl package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Frank Schäfer, Michal Kloc, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;1&lt;/strong&gt;, 035009 (2020).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Frank Schäfer, Pavel Sekatski, et al., Mach. Learn.: Sci. Technol. &lt;strong&gt;2&lt;/strong&gt;, 035004 (2021).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Chris Rackauckas, Yingbo Ma, et al., arXiv preprint arXiv:2001.04385 (2020).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Qiqi Wang, Rui Hu, et al. J. Comput. Phys &lt;strong&gt;26&lt;/strong&gt;, 210-224 (2014)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:5&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017). &lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:5&#34;&gt;
+&lt;p&gt;Angxiu Ni and Qiqi Wang. J. Comput. Phys &lt;strong&gt;347&lt;/strong&gt;,  56-77 (2017).&amp;#160;&lt;a href=&#34;#fnref:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref4:5&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:6&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019) &lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:6&#34;&gt;
+&lt;p&gt;Angxiu Ni and Chaitanya Talnikar, J. Comput. Phys &lt;strong&gt;395&lt;/strong&gt;, 690-709, (2019)&amp;#160;&lt;a href=&#34;#fnref:6&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:7&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019) &lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:7&#34;&gt;
+&lt;p&gt;Angxiu Ni, Qiqi Wang et al., J. Comput. Phys &lt;strong&gt;394&lt;/strong&gt;, 615-631 (2019)&amp;#160;&lt;a href=&#34;#fnref:7&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
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+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
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+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
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+              <span>Light</span>
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+              <span>Automatic</span>
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+    
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+    
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+
+
+
+
+
+
+
+
+
+
+  
+
+  
+  
+  
+    
+  
+<div class="universal-wrapper pt-3">
+  <h1>Source Themes</h1>
+
+  
+
+  
+</div>
+
+
+
+<div class="universal-wrapper">
+  
+
+
+  
+
+  
+  
+    
+
+
+
+
+
+
+
+  
+
+
+
+
+
+
+
+  
+
+
+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/publication/preprint/" >An example preprint / working paper</a>
+    </div>
+
+    
+    <a href="/publication/preprint/"  class="summary-link">
+      <div class="article-style">
+        Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+      <div>
+        
+
+  <span class="author-highlighted">
+      Frank Schäfer</span>
+      </div>
+      
+    </div>
+
+    
+    <div class="btn-links">
+      
+
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+
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+  
+    
+  
+
+
+
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="http://arxiv.org/pdf/1512.04133v1" target="_blank" rel="noopener">
+  PDF
+</a>
+
+
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
+  Code
+</a>
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="/#" target="_blank" rel="noopener">
+  Dataset
+</a>
+
+
+
+  
+    
+  
+
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="/#" target="_blank" rel="noopener">
+  Poster
+</a>
+
+
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="/slides/example/" target="_blank">
+    Slides
+  </a>
+  
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="/#" target="_blank" rel="noopener">
+  Video
+</a>
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="/#" target="_blank" rel="noopener">
+  Source Document
+</a>
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+
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="http://example.org" target="_blank" rel="noopener">
+    Custom Link</a>
+
+
+    </div>
+    
+
+  </div>
+  <div class="ml-3">
+    
+    
+      
+      
+      <a href="/publication/preprint/" >
+        <img src="/publication/preprint/featured_hu559a5add5185b02575aa8333502ab2cc_220813_46db05eae868715a20e63a4c49a84677.webp" height="100" width="150"
+            alt="An example preprint / working paper" loading="lazy">
+      </a>
+    
+  </div>
+</div>
+
+  
+    
+
+
+
+
+
+
+
+  
+
+
+
+
+
+
+
+  
+
+
+<div class="media stream-item view-compact">
+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/publication/journal-article/" >An example journal article</a>
+    </div>
+
+    
+    <a href="/publication/journal-article/"  class="summary-link">
+      <div class="article-style">
+        Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellus ac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum.
+      </div>
+    </a>
+    
+
+    <div class="stream-meta article-metadata">
+
+      
+
+      
+      <div>
+        
+
+  <span class="author-highlighted">
+      Frank Schäfer</span><i class="author-notes fas fa-info-circle" data-toggle="tooltip" title="Equal contribution"></i>, <span >
+      Robert Ford</span><i class="author-notes fas fa-info-circle" data-toggle="tooltip" title="Equal contribution"></i>
+      </div>
+      
+    </div>
+
+    
+    <div class="btn-links">
+      
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+  
+    
+  
+
+
+
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="http://arxiv.org/pdf/1512.04133v1" target="_blank" rel="noopener">
+  PDF
+</a>
+
+
+
+<a href="#" class="btn btn-outline-primary btn-page-header btn-sm js-cite-modal"
+        data-filename="/publication/journal-article/cite.bib">
+  Cite
+</a>
+
+
+  
+  
+    
+  
+<a class="btn btn-outline-primary btn-page-header btn-sm" href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">
+  Code
+</a>
+
+
+
+
+
+
+
+  
+  <a class="btn btn-outline-primary btn-page-header btn-sm" href="/slides/example/" target="_blank">
+    Slides
+  </a>
+  
+
+
+
+
+
+
+    </div>
+    
+
+  </div>
+  <div class="ml-3">
+    
+    
+      
+      
+      <a href="/publication/journal-article/" >
+        <img src="/publication/journal-article/featured_hu3d03a01dcc18bc5be0e67db3d8d209a6_214168_657db23a05c47b2ca698a87473c22df8.webp" height="91" width="150"
+            alt="An example journal article" loading="lazy">
+      </a>
+    
+  </div>
+</div>
+
+  
+
+  
+
+  
+
+</div>
+  </div>
+
+  <div class="page-footer">
+    
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+    <div class="container">
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+
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+
+
+
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+
+
+
+  
+  
+  
+
+  
+  
+    
+  
+  
+    
+  
+
+  
+
+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
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+<script src="/js/vendor-bundle.min.ff7771056d34ad9f2de2d8f6a466e748.js"></script>
+
+
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+  
+
+  
+  
+
+  
+
+
+
+
+
+
+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
+
+
+
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+
+
+  
+  <script id="search-hit-fuse-template" type="text/x-template">
+    <div class="search-hit" id="summary-{{key}}">
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+        <div class="search-hit-name">
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+    <script src="https://cdn.jsdelivr.net/gh/julmot/mark.js@8.11.1/dist/jquery.mark.min.js" integrity="sha512-mhbv5DqBMgrWL+32MmsDOt/OAvqr/cHimk6B8y/bx/xS88MVkYGPiVv2ixKVrkywF2qHplNRUvFsAHUdxZ3Krg==" crossorigin="anonymous"></script>
+  
+
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+  
+  
+  
+  
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+  
+  
+
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
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+
+
+  
+<div id="modal" class="modal fade" role="dialog">
+  <div class="modal-dialog">
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+        <h5 class="modal-title">Cite</h5>
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+
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+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
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diff --git a/tag/source-themes/index.xml b/tag/source-themes/index.xml
new file mode 100644
index 0000000..6163dff
--- /dev/null
+++ b/tag/source-themes/index.xml
@@ -0,0 +1,49 @@
+<?xml version="1.0" encoding="utf-8" standalone="yes" ?>
+<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
+  <channel>
+    <title>Source Themes | FS</title>
+    <link>https://frankschae.github.io/tag/source-themes/</link>
+      <atom:link href="https://frankschae.github.io/tag/source-themes/index.xml" rel="self" type="application/rss+xml" />
+    <description>Source Themes</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sun, 07 Apr 2019 00:00:00 +0000</lastBuildDate>
+    <image>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Source Themes</title>
+      <link>https://frankschae.github.io/tag/source-themes/</link>
+    </image>
+    
+    <item>
+      <title>An example preprint / working paper</title>
+      <link>https://frankschae.github.io/publication/preprint/</link>
+      <pubDate>Sun, 07 Apr 2019 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/preprint/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
+</description>
+    </item>
+    
+    <item>
+      <title>An example journal article</title>
+      <link>https://frankschae.github.io/publication/journal-article/</link>
+      <pubDate>Tue, 01 Sep 2015 00:00:00 +0000</pubDate>
+      <guid>https://frankschae.github.io/publication/journal-article/</guid>
+      <description>&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Click the &lt;em&gt;Cite&lt;/em&gt; button above to demo the feature to enable visitors to import publication metadata into their reference management software.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;div class=&#34;alert alert-note&#34;&gt;
+  &lt;div&gt;
+    Create your slides in Markdown - click the &lt;em&gt;Slides&lt;/em&gt; button to check out the example.
+  &lt;/div&gt;
+&lt;/div&gt;
+&lt;p&gt;Add the publication&amp;rsquo;s &lt;strong&gt;full text&lt;/strong&gt; or &lt;strong&gt;supplementary notes&lt;/strong&gt; here. You can use rich formatting such as including &lt;a href=&#34;https://docs.hugoblox.com/content/writing-markdown-latex/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;code, math, and images&lt;/a&gt;.&lt;/p&gt;
+</description>
+    </item>
+    
+  </channel>
+</rss>
diff --git a/tag/source-themes/page/1/index.html b/tag/source-themes/page/1/index.html
new file mode 100644
index 0000000..be04af0
--- /dev/null
+++ b/tag/source-themes/page/1/index.html
@@ -0,0 +1,10 @@
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/tag/source-themes/</title>
+    <link rel="canonical" href="https://frankschae.github.io/tag/source-themes/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/source-themes/">
+  </head>
+</html>
diff --git a/tag/srk-methods/index.html b/tag/srk-methods/index.html
index 38be96c..2cf30fd 100644
--- a/tag/srk-methods/index.html
+++ b/tag/srk-methods/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tag/srk-methods/">
+
+
+
+
+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
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-  <h1>SRK methods</h1>
+  <h1>SRK Methods</h1>
 
   
 
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 <div class="universal-wrapper">
   
 
+
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+  <div class="media-body">
+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/high-weak/" >High weak order SDE solvers</a>
+    </div>
+
     
-    
-    <div>
-      <h2><a href="/post/high-weak/" >High weak order SDE solvers</a></h2>
+    <a href="/post/high-weak/"  class="summary-link">
       <div class="article-style">
-        
-          This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
+        This post summarizes our new high weak order methods for the SciML ecosystem, as implemented within the Google Summer of Code 2020 project.
 After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
-        
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-    </div>
+    </a>
+    
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+    <div class="stream-meta article-metadata">
+
+      
+
+      
+        
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+<div class="article-metadata">
+
+  
+  
+
   
+  <span class="article-date">
     
     
+      
+          Last updated on
+      
     
-    <div>
-      <h2><a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a></h2>
-      <div class="article-style">
-        
-          First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
-        
-      </div>
-    </div>
+    Aug 18, 2020
+  </span>
   
 
   
 
+  
+  <span class="middot-divider"></span>
+  <span class="article-reading-time">
+    10 min read
+  </span>
+  
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+
+    <div class="section-subheading article-title mb-0 mt-0">
+      <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/" >GSoC 2020: High weak order SDE solvers and their utility in neural SDEs</a>
+    </div>
+
+    
+    <a href="/post/gsoc2020-high-weak-order-solvers-sde-adjoints/"  class="summary-link">
+      <div class="article-style">
+        First and foremost, I would like to thank my mentors Chris Rackauckas, Moritz Schauer, and Yingbo Ma for their willingness to supervise me in this Google Summer of Code project. Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion.
+      </div>
+    </a>
+    
+
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diff --git a/tag/srk-methods/index.xml b/tag/srk-methods/index.xml
index 09db4c3..9910cb6 100644
--- a/tag/srk-methods/index.xml
+++ b/tag/srk-methods/index.xml
@@ -1,14 +1,14 @@
 <?xml version="1.0" encoding="utf-8" standalone="yes" ?>
 <rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
   <channel>
-    <title>SRK methods | FS</title>
+    <title>SRK Methods | FS</title>
     <link>https://frankschae.github.io/tag/srk-methods/</link>
       <atom:link href="https://frankschae.github.io/tag/srk-methods/index.xml" rel="self" type="application/rss+xml" />
-    <description>SRK methods</description>
-    <generator>Source Themes Academic (https://sourcethemes.com/academic/)</generator><language>en-us</language><copyright>© 2020 Frank Schäfer</copyright><lastBuildDate>Mon, 17 Aug 2020 14:46:35 +0200</lastBuildDate>
+    <description>SRK Methods</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Mon, 17 Aug 2020 14:46:35 +0200</lastBuildDate>
     <image>
-      <url>https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png</url>
-      <title>SRK methods</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>SRK Methods</title>
       <link>https://frankschae.github.io/tag/srk-methods/</link>
     </image>
     
@@ -17,11 +17,8 @@
       <link>https://frankschae.github.io/post/high-weak/</link>
       <pubDate>Mon, 17 Aug 2020 14:46:35 +0200</pubDate>
       <guid>https://frankschae.github.io/post/high-weak/</guid>
-      <description>&lt;p&gt;This post summarizes our new 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
-for the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the 
-&lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
+      <description>&lt;p&gt;This post summarizes our new &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;high weak order methods&lt;/a&gt;
+for the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; ecosystem, as implemented within the &lt;a href=&#34;https://summerofcode.withgoogle.com/organizations/6363760870031360/?sp-page=2#5505348691034112&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Google Summer of Code 2020&lt;/a&gt; project.&lt;/p&gt;
 &lt;p&gt;After an introductory part highlighting the differences between the strong and the weak approximation for stochastic differential equations, we look into the convergence and performance properties of a few representative new methods in case of a non-commutative noise process.
 Based on the stochastic version of the Brusselator equations, we demonstrate how adaptive step-size control for the weak solvers can result in a better approximation of the system dynamics.
 Finally, we discuss how to run simulations on GPU hardware.&lt;/p&gt;
@@ -52,16 +49,14 @@ $$
 \rm{E}(|X(T) -Y(T)|) \leq C \cdot h^p,
 $$
 for each $h \in [0, \delta]$.&lt;/p&gt;
-&lt;p&gt;The 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
+&lt;p&gt;The &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains various state-of-the-art solvers
 for the strong approximation of SDEs. In most cases, the strong solvers are however restricted to special noise forms.
 For example, the very powerful stability-optimized, adaptive strong order 3/2 stochastic Runge-Kutta method (SOSRI)
 can only handle diagonal and scalar noise Ito SDEs, i.e., noise processes where &lt;em&gt;b&lt;/em&gt; has only entries on its diagonal or $m=1$.
 The main difficulty for the construction of strong methods with an order &amp;gt; 1/2 arises from the need of an accurate estimation of
 multiple stochastic integrals. While the iterated stochastic integrals can be expressed in terms of &lt;em&gt;dW&lt;/em&gt; in the case
 of scalar, diagonal, and commutative noise processes, an approximation based on a Fourier expansion of a Brownian bridge must be employed
-in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also 
-&lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
+in the case of non-commutative noise processes.&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; Currently, we are also &lt;a href=&#34;https:://github.com/SciML/StochasticDiffEq.jl/pull/347&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;implementing those iterated integrals in the StochasticDiffEq library&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;weak-convergence&#34;&gt;Weak convergence&lt;/h2&gt;
 &lt;p&gt;Instead of an accurate pathwise approximation of a stochastic process, we only require an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt; in many situations. In these cases, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient and &amp;ndash; due to the less restrictive formulation of the objective &amp;ndash; those solvers are computationally cheaper than their strong counterparts.
 For example, weak solvers are very efficient for simulations in quantum optics, if only mean values of many trajectories are required, e.g., when the expectation values of variables such as position and momentum operators are computed in the phase space framework (Wigner functions, positive P-functions, etc.) of quantum mechanics.
@@ -76,19 +71,17 @@ $~$&lt;/p&gt;
 &lt;p&gt;for any polynomial $g$ for each $h \in [0, \delta]$.&lt;/p&gt;
 &lt;p&gt;We demonstrate below that &lt;strong&gt;high weak order solvers&lt;/strong&gt; are specifically appealing, as they allow for using much larger time steps while attaining the same error in the mean, as compared with SDE solvers possessing a smaller weak-order convergence.&lt;/p&gt;
 &lt;h2 id=&#34;new-high-weak-order-methods&#34;&gt;New high weak order methods&lt;/h2&gt;
-&lt;p&gt;A list of all new weak solvers is available in the 
-&lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
+&lt;p&gt;A list of all new weak solvers is available in the &lt;a href=&#34;https://diffeq.sciml.ai/dev/solvers/sde_solve/#High-Weak-Order-Methods-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML documentation&lt;/a&gt;.
 Note that we also implemented methods designed for the Stratonovich sense.
 For the subsequent examples regarding Ito SDEs, we use only a subset of the plethora of second-order weak solvers.
-We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
-We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
+We employ the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;, &lt;code&gt;RD1WM()&lt;/code&gt;&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;, and &lt;code&gt;RD2WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; methods due to Debrabant &amp;amp; Rößler and Platen&amp;rsquo;s &lt;code&gt;PL1WM()&lt;/code&gt;&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; method.
+We compare those methods to the strong Euler-Maruyama &lt;code&gt;EM()&lt;/code&gt;&lt;sup id=&#34;fnref2:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and the simplified Euler-Maruyama &lt;code&gt;SimplifiedEM()&lt;/code&gt;&lt;sup id=&#34;fnref3:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; schemes.
 The latter is the simplest weak solver, where the Gaussian increments of the strong Euler-Maruyama scheme are replaced by
 two-point distributed random variables with similar moment properties.&lt;/p&gt;
 &lt;p&gt;Rößler&amp;rsquo;s SRK schemes are particularly designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;, since only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn and since the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.
 &lt;code&gt;PL1WM()&lt;/code&gt; in contrast needs to simulate &lt;code&gt;m(m+1)/2&lt;/code&gt; random variables but a smaller number of order conditions needs to be fulfilled.&lt;/p&gt;
 &lt;h3 id=&#34;convergence-tests&#34;&gt;Convergence tests&lt;/h3&gt;
-&lt;p&gt;As in the 
-&lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As in the &lt;a href=&#34;https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;first blog post&lt;/a&gt;, let us consider the multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -99,63 +92,61 @@ $$ ~$$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;for the function $f(x)=(x_1)^2$, which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e6&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-using Test
-using Random
-using Plots
-using DiffEqDevTools
-
-function prob_func(prob, i, repeat)
-    remake(prob,seed=seeds[i])
-end
-
-u₀ = [1.0,1.0]
-function f1!(du,u,p,t)
-  @inbounds begin
-    du[1] = -273//512*u[1]
-    du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-  end
-  return nothing
-end
-function g1!(du,u,p,t)
-  @inbounds begin
-    du[1,1] = 1//4*u[1]
-    du[1,2] = 1//16*u[1]
-    du[2,1] = (1-2*sqrt(2))/4*u[1]
-    du[2,2] = 1//10*u[1]+1//16*u[2]
-  end
-  return nothing
-end
-dts = 1 .//2 .^(3: -1:0)
-tspan = (0.0,3.0)
-
-h2(z) = z^2 # but apply it only to u[1]
-
-prob = SDEProblem(f1!,g1!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-numtraj = Int(1e6)
-seed = 100
-Random.seed!(seed)
-seeds = rand(UInt, numtraj)
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h2(sol[end][1]),false),
-        prob_func = prob_func
-        )
-
-sim = test_convergence(dts,ensemble_prob,DRI1(),
-  save_everystep=false,trajectories=numtraj,save_start=false,adaptive=false,
-  weak_timeseries_errors=false,weak_dense_errors=false,
-  expected_value=exp(-3.0)
-  )
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqDevTools&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;return&lt;/span&gt; &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;3&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# but apply it only to u[1]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g1!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Int&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;test_convergence&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The object &lt;code&gt;sim&lt;/code&gt; defined in the last line contains all relevant quantities to test the weak convergence with respect to the final time point for the &lt;code&gt;DRI1()&lt;/code&gt; scheme.
 Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function for the other aforementioned solvers, we obtain the convergence plot:&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -167,18 +158,12 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/weak_conv.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/weak_conv.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;weak_conv.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Note that the &lt;code&gt;SimplifiedEM&lt;/code&gt; and the &lt;code&gt;EM&lt;/code&gt; scheme fall on top of each other.
 &lt;code&gt;DRI1()&lt;/code&gt; achieves the smallest errors for a fixed &lt;code&gt;dt&lt;/code&gt; in this study.&lt;/p&gt;
@@ -186,35 +171,29 @@ Repeating this call to the &lt;code&gt;test_convergence()&lt;/code&gt; function
 &lt;p&gt;Ultimately, we are not only interested in the general convergence slope of an algorithm but also in its speed.
 We&amp;rsquo;d like to select the fastest method depending on the permitted tolerance.
 This is commonly studied using work-precision diagram.
-Thanks to 
-&lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;reltols = 1.0 ./ 4.0 .^ (1:4)
-abstols = reltols#[0.0 for i in eachindex(reltols)]
-setups = [
-          Dict(:alg=&amp;gt;DRI1(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;PL1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;EM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;SimplifiedEM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI2WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false),
-          Dict(:alg=&amp;gt;RDI1WM(),:dts=&amp;gt;dts,:adaptive=&amp;gt;false)
-          ]
-test_dt = 1//10000
-appxsol_setup = Dict(:alg=&amp;gt;EM(), :dt=&amp;gt;test_dt)
-wp = @time WorkPrecisionSet(ensemble_prob,
-                            abstols,reltols,setups,test_dt;
-                            maxiters = 1e7,verbose=false,
-                            save_everystep=false,save_start=false,
-                            appxsol_setup = appxsol_setup,
-                            expected_value=exp(-3.0),
-                            trajectories=numtraj, error_estimate=:weak_final)
-plt = plot(wp;legend=:bottomleft)
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
-
+Thanks to &lt;a href=&#34;https://github.com/SciML/DiffEqDevTools.jl/pull/75&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;new routines&lt;/a&gt;, a user can generate a work-precision diagram by the following code&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;./&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;4.0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;c&#34;&gt;#[0.0 for i in eachindex(reltols)]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;PL1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;SimplifiedEM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI2WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RDI1WM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:dts&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;          &lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10000&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;kt&#34;&gt;Dict&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:alg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EM&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt; &lt;span class=&#34;ss&#34;&gt;:dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&amp;gt;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WorkPrecisionSet&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;abstols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reltols&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;setups&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;test_dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;maxiters&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;verbose&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;appxsol_setup&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;expected_value&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;3.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;                            &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;error_estimate&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:weak_final&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;wp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:bottomleft&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
-  
 
 
 
@@ -226,88 +205,87 @@ plt = plot(wp;legend=:bottomleft)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/WorkPrecision.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;WorkPrecision.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;Therefore, &lt;code&gt;DRI1&lt;/code&gt; has the best performance in this non-commutative noise case if the error is supposed to stay below 1e-3.
 For larger permitted errors, the &lt;code&gt;SimplifiedEM&lt;/code&gt; scheme might be a good choice. However, the first order methods
 are outclassed when high precision is more of a concern.
-We plan to perform more in-depth benchmarks in the near future what will be reported on the 
-&lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
+We plan to perform more in-depth benchmarks in the near future what will be reported on the &lt;a href=&#34;https://sciml.ai/news/2020/08/10/StochasticBonanza/#tons_of_methods_for_high_weak_order_solving_of_sdes&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML news&lt;/a&gt;. Stay tuned!&lt;/p&gt;
 &lt;h3 id=&#34;adaptive-step-size-control&#34;&gt;Adaptive step-size control&lt;/h3&gt;
-&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
+&lt;p&gt;Already in 2004, Rößler proposed an adaptive discretization algorithm for the weak approximation of SDEs.&lt;sup id=&#34;fnref2:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; The idea is to employ an embedded SRK scheme: Using the same function evaluations but distinct Butcher tableaus, one constructs two stochastic Runge-Kutta methods with different convergence order, such that the local error can be estimated with only small additional computational overhead. Based on the error estimate, new step sizes are proposed.&lt;/p&gt;
 &lt;p&gt;To use adaptive step-size control, it is sufficient to set &lt;code&gt;adaptive=true&lt;/code&gt; (default setting). Optionally, one may also pass absolute and relative tolerances.&lt;/p&gt;
 &lt;p&gt;The following julia code simulates the stochastic version of the Brusselator equations with intitial condition&lt;/p&gt;
 &lt;p&gt;$$ X(t=0)=  \begin{pmatrix} 0.1 \\ 0\end{pmatrix},$$&lt;/p&gt;
 &lt;p&gt;on a time span $\mathbb{I}=[0, 100]$ for adaptive (&lt;code&gt;sol&lt;/code&gt;) and fixed step sizes (&lt;code&gt;sol_na&lt;/code&gt;):&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq, DiffEqNoiseProcess, Random
-using Plots
-using DiffEqGPU
-
-function prob_func(prob, i, repeat)
-  Random.seed!(seeds[i])
-  W = WienerProcess(0.0,0.0,0.0)
-  remake(prob,noise=W)
-end
-
-function brusselator_f!(du,u,p,t)
-  @inbounds begin
-    du[1] = (p[1]-1)*u[1]+p[1]*u[1]^2+(u[1]+1)^2*u[2]
-    du[2] = -p[1]*u[1]-p[1]*u[1]^2-(u[1]+1)^2*u[2]
-  end
-  nothing
-end
-
-function scalar_noise!(du,u,p,t)
-  @inbounds begin
-    du[1] = p[2]*u[1]*(1+u[1])
-    du[2] = -p[2]*u[1]*(1+u[1])
-   end
-   nothing
-end
-
-
-# fix seeds
-seed = 100
-Random.seed!(seed)
-numtraj= 100
-seeds = rand(UInt, numtraj)
-W = WienerProcess(0.0,0.0,0.0)
-
-
-u0 = [-0.1f0,0.0f0]
-tspan = (0.0f0,100.0f0)
-p = [1.9f0,0.1f0]
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Plots&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqGPU&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;repeat&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;remake&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;   &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# fix seeds&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seeds&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rand&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;kt&#34;&gt;UInt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;WienerProcess&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100.0f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.9f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;brusselator_f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;scalar_noise!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;prob_func&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@time&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensembleprob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleCPUArray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleSummary&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5f0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;100f0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;pl&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;summ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;fillalpha&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;time t&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;X(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
-prob = SDEProblem(brusselator_f!,scalar_noise!,u0,tspan,p,noise=W)
 
-ensembleprob = EnsembleProblem(prob, prob_func = prob_func)
 
-sol = @time solve(ensembleprob,DRI1(),dt=0.1,EnsembleCPUArray(),trajectories=numtraj)
-sol_na = @time solve(ensembleprob,DRI1(),dt=0.8,adaptive=false,EnsembleCPUArray(),trajectories=numtraj)
 
 
-summ = EnsembleSummary(sol,0.0f0:0.5f0:100f0)
-pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quot;X(t)&amp;quot;, label= [&amp;quot;x₁(t)&amp;quot; &amp;quot;x₂(t)&amp;quot;], legend=true)
 
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;The time evolution of both dependent variables ($x_1(t)$ and $x_2(t)$) displays damped oscillations.&lt;/p&gt;
 
 
 
 
 
-  
 
 
 
@@ -315,72 +293,55 @@ pl = plot(summ,fillalpha=0.5,xlabel = &amp;quot;time t&amp;quot;, yaxis=&amp;quo
 
 
 
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator_many_trajectories.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
-
-
-
-&lt;figure &gt;
-
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator_many_trajectories.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
-
-&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
+&lt;p&gt;We can confirm Rößler&amp;rsquo;s observation&lt;sup id=&#34;fnref3:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; that the adaptive scheme describes the time evolution of the SDE more accurately,
 as oscillations are damped out stronger for the fixed step size method, thus approaching the origin too rapidly.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using DifferentialEquations.EnsembleAnalysis
-meansol = timeseries_steps_mean(sol)
-meansol_na = timeseries_point_mean(sol_na,meansol.t)
-dts = []
-tmp1 = tspan[1]
-for tmp2 in meansol.t
-  global tmp1
-  push!(dts,tmp2-tmp1)
-  tmp1 = tmp2
-end
-#
-
-list_plots = []
-for i in 1:length(meansol.u)
-  l = @layout [a  b]
-  plt1 = plot(meansol[1, 1:i],meansol[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;adaptive&amp;quot;,
-         lw=2,
-         linecolor=1
-         )
-  plot!(meansol_na[1, 1:i],meansol_na[2, 1:i],
-         ylim = (-0.18, 0.18),
-         xlim = (-0.13, 0.13),
-         xlabel = &amp;quot;x₁(t)&amp;quot;,
-         yaxis= &amp;quot;x₂(t)&amp;quot;,
-         label=&amp;quot;fixed step size&amp;quot;,
-         lw=2,
-         linecolor=2
-         )
-  pl2 = scatter(dts[1:i], xlabel = &amp;quot;step&amp;quot;, yaxis= &amp;quot;dtᵢ&amp;quot;, xlim = (0, length(meansol.u)),  ylim = (0.0, 2.3), legend=false)
-  plt = plot(plt1, pl2, layout = l)
-  push!(list_plots, plt)
-end
-
-anim = animate(list_plots,lw=2,every=1)
-
-&lt;/code&gt;&lt;/pre&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DifferentialEquations&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EnsembleAnalysis&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_steps_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;timeseries_point_mean&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;global&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tmp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tmp2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;#&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;i&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;nd&#34;&gt;@layout&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;a&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;b&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;adaptive&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plot!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol_na&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.18&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.13&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₁(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;x₂(t)&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;label&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;s&#34;&gt;&amp;#34;fixed step size&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;n&#34;&gt;linecolor&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;         &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;scatter&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dts&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlabel&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;step&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;yaxis&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;s&#34;&gt;&amp;#34;dtᵢ&amp;#34;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;xlim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;length&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;meansol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)),&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;ylim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;2.3&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;legend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plot&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;plt1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;pl2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;layout&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;l&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;push!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;plt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;anim&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;animate&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;list_plots&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;lw&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;every&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
-  
 
 
 
@@ -392,53 +353,46 @@ anim = animate(list_plots,lw=2,every=1)
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/Brusselator.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;Brusselator.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gpu-usage&#34;&gt;GPU usage&lt;/h3&gt;
 &lt;p&gt;All necessary tools to accelerate the simulation of (stochastic) differential equations on GPUs within the SciML
-ecosystem are collected in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
+ecosystem are collected in the &lt;a href=&#34;https://github.com/SciML/DiffEqGPU.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqGPU&lt;/a&gt; package.&lt;/p&gt;
 &lt;p&gt;Currently, bounds checking and return values are not allowed, i.e., functions must be
 written in the form:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;function f!(du,u,p,t)
-  @inbounds begin
-    du[1] = ..
-  end
-  nothing
-end
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nd&#34;&gt;@inbounds&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;begin&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;    &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;..&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;nb&#34;&gt;nothing&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Except from those limitations, a user can specifiy &lt;code&gt;ensemblealg=EnsembleGPUArray()&lt;/code&gt; to parallelize SDE solves across the GPU, see, e.g., &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/blob/master/test/gpu/sde_weak_adaptive.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;the GPU tests for StochasticDiffEq&lt;/a&gt; for some examples.
 Note that for some high weak order solvers GPU usage is not recommended as scalar indexing is used.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations. &lt;strong&gt;23&lt;/strong&gt;, Springer Science &amp;amp; Business Media (2013).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Peter E. Kloeden, Eckhard Platen, and Ian W. Wright, Stochastic analysis and applications &lt;strong&gt;10&lt;/strong&gt; 431-441 (1992).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008) &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Mathematics and Computers in Simulation &lt;strong&gt;77&lt;/strong&gt;, 408-420 (2008)&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref2:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref3:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
@@ -447,21 +401,14 @@ Note that for some high weak order solvers GPU usage is not recommended as scala
       <link>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</link>
       <pubDate>Sat, 30 May 2020 15:10:33 +0200</pubDate>
       <guid>https://frankschae.github.io/post/gsoc2020-high-weak-order-solvers-sde-adjoints/</guid>
-      <description>&lt;p&gt;First and foremost, I would like to thank my mentors 
-&lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, 
-&lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  
-&lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
+      <description>&lt;p&gt;First and foremost, I would like to thank my mentors &lt;a href=&#34;https://github.com/ChrisRackauckas&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Chris Rackauckas&lt;/a&gt;, &lt;a href=&#34;https://github.com/mschauer&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Moritz Schauer&lt;/a&gt;, and  &lt;a href=&#34;https://github.com/YingboMa&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;Yingbo Ma&lt;/a&gt; for their willingness to supervise me in this Google Summer of Code project.
 Although we are still at the very beginning of the project, we already had plenty of very inspiring discussion. I will spend the following months implementing both  &lt;strong&gt;new high weak order solvers&lt;/strong&gt; as well as &lt;strong&gt;adjoint sensitivity methods&lt;/strong&gt; for stochastic differential equations (SDEs).
-The project is embedded within the 
-&lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
-Ultimately, the planned contributions will allow researchers to simulate (or even 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where 
-&lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
+The project is embedded within the &lt;a href=&#34;https://sciml.ai&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SciML&lt;/a&gt; organization which, among others, unifies the latest toolsets from scientific machine learning and differential equation solver software.
+Ultimately, the planned contributions will allow researchers to simulate (or even &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/LV-stochastic/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;control&lt;/a&gt;) stochastic dynamics. Also inverse problems, where &lt;a href=&#34;https://diffeqflux.sciml.ai/dev/examples/NN-SDE/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;SDE models are fit to data&lt;/a&gt;, fall into the scope.
 Therefore, relevant applications are found in many fields ranging from the simulation of (bio-)chemical processes over financial modeling to quantum mechanics.&lt;/p&gt;
 &lt;p&gt;This post is supposed to summarize what we have implemented in this first period and what we are going to do next. Future posts are going to dig into the individual subjects in more details.&lt;/p&gt;
 &lt;h2 id=&#34;high-weak-order-solvers&#34;&gt;High Weak Order Solvers&lt;/h2&gt;
-&lt;p&gt;Currently, the 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
+&lt;p&gt;Currently, the &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;StochasticDiffEq&lt;/a&gt; package contains state-of-the-art solvers for the strong approximation of SDEs, i.e., solvers that allow one to reconstruct correctly the numerical solution of an SDE in a pathwise sense.
 In general, an accurate estimation of multiple stochastic integrals is then required to produce a strong method of order greater than 1/2.&lt;/p&gt;
 &lt;p&gt;However in many situations, we are only aiming for computing an estimation for the &lt;strong&gt;expected value of the solution&lt;/strong&gt;.
 In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&gt; are sufficient. The less restrictive formulation of the objective for weak methods has the advantage that they are computationally cheaper than strong methods.
@@ -469,7 +416,7 @@ In such situations, methods for the &lt;strong&gt;weak approximation&lt;/strong&
 As an example, when Monte Carlo methods are used for SDE models, it is indeed often sufficient to be able to accurately sample random trajectories of the SDE, and it is not important to accurately approximate a particular trajectory. The former is exactly what a solver with high weak order provides.&lt;/p&gt;
 &lt;h3 id=&#34;second-order-runge-kutta-methods-for-ito-sdes&#34;&gt;Second order Runge-Kutta methods for Ito SDEs&lt;/h3&gt;
 &lt;p&gt;In the beginning of the community bonding period I finished the implementations of the &lt;code&gt;DRI1()&lt;/code&gt;&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt; and &lt;code&gt;RI1()&lt;/code&gt;&lt;sup id=&#34;fnref:2&#34;&gt;&lt;a href=&#34;#fn:2&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;2&lt;/a&gt;&lt;/sup&gt; methods. Both are representing second order Runge-Kutta schemes and were introduced by Rößler. Interestingly, these methods are designed to scale well with the number of Wiener processes &lt;code&gt;m&lt;/code&gt;. Specifically, only &lt;code&gt;2m-1&lt;/code&gt; random variables have to be drawn (in contrast to &lt;code&gt;m(m+1)/2&lt;/code&gt; from previous methods). Additionally, the number of function evaluations for the drift and the diffusion terms is independent of &lt;code&gt;m&lt;/code&gt;.&lt;/p&gt;
-&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
+&lt;p&gt;As an example, we can check the second order convergence property on a multi-dimensional SDE with non-commuting noise&lt;sup id=&#34;fnref1:1&#34;&gt;&lt;a href=&#34;#fn:1&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;1&lt;/a&gt;&lt;/sup&gt;:&lt;/p&gt;
 &lt;p&gt;$$
 \scriptstyle d \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} = \begin{pmatrix} -\frac{273}{512} &amp;amp;  \phantom{X_2}0 \\  -\frac{1}{160} \phantom{X_2}  &amp;amp; -\frac{785}{512}+\frac{\sqrt{2}}{8} \end{pmatrix}  \begin{pmatrix} X_1 \\  X_2 \end{pmatrix} dt + \begin{pmatrix} \frac{1}{4} X_1 &amp;amp;  \frac{1}{16} X_1 \\  \frac{1-2\sqrt{2}}{4} X_2 &amp;amp; \frac{1}{10}X_1  +\frac{1}{16} X_2 \end{pmatrix} d \begin{pmatrix} W_1 \\  W_2 \end{pmatrix} &lt;br&gt;
 $$&lt;/p&gt;
@@ -478,47 +425,40 @@ $$&lt;/p&gt;
 &lt;p&gt;$$ \rm{E}\left[ f(X(t)) \right] =  \exp(-t),$$&lt;/p&gt;
 &lt;p&gt;which we use to test the weak convergence order of the algorithms in the following.&lt;/p&gt;
 &lt;p&gt;To compute the expected value numerically, we sample an ensemble of &lt;code&gt;numtraj = 1e7&lt;/code&gt; trajectories for different step sizes &lt;code&gt;dt&lt;/code&gt;. The code for a single  &lt;code&gt;dt&lt;/code&gt; reads:&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using StochasticDiffEq
-numtraj = 1e7
-u₀ = [1.0,1.0]
-function f!(du,u,p,t)
-  du[1] = -273//512*u[1]
-  du[2] = -1//160*u[1]-(-785//512+sqrt(2)/8)*u[2]
-end
-function g!(du,u,p,t)
-  du[1,1] = 1//4*u[1]
-  du[1,2] = 1//16*u[1]
-  du[2,1] = (1-2*sqrt(2))/4*u[1]
-  du[2,2] = 1//10*u[1]+1//16*u[2]
-end
-dt = 1//8
-tspan = (0.0,10.0)
-prob = SDEProblem(f!,g!,u₀,tspan,noise_rate_prototype=zeros(2,2))
-
-h(z) = z^2
-
-ensemble_prob = EnsembleProblem(prob;
-        output_func = (sol,i) -&amp;gt; (h(sol[end][1]),false)
-        )
-sol = solve(ensemble_prob, DRI1();
-        dt=dt,
-        save_start=false,
-        save_everystep=false,
-        weak_timeseries_errors=false,
-        weak_dense_errors=false,
-        trajectories=numtraj)
-
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
-&lt;pre&gt;&lt;code&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
-
-
-
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;273&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;160&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;785&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;512&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;-&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sqrt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;4&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;10&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;16&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;//&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;8&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;10.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise_rate_prototype&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;zeros&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;z&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;EnsembleProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;;&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;output_func&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;-&amp;gt;&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;h&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]),&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;ensemble_prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DRI1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;();&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_start&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;save_everystep&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_timeseries_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;weak_dense_errors&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;        &lt;span class=&#34;n&#34;&gt;trajectories&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;numtraj&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;We then compute the error of the numerically obtained expected value of the ensemble simulation with respect to the analytical result:&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-fallback&#34; data-lang=&#34;fallback&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;LinearAlgebra.norm(Statistics.mean(sol.u)-exp(-tspan[2]))
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;Repeating this procedure for some more values of &lt;code&gt;dt&lt;/code&gt;, the log-log plot of the error as a function of &lt;code&gt;dt&lt;/code&gt; displays nicely the second order convergence (slope $\approx 2.2$).&lt;/p&gt;
 
 
-  
 
 
 
@@ -530,29 +470,24 @@ sol = solve(ensemble_prob, DRI1();
 
 
 
-&lt;figure &gt;
 
 
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/DRI1.png&#34; &gt;
 
 
-  &lt;img src=&#34;https://frankschae.github.io/img/DRI1.png&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
 
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;DRI1.png&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;p&gt;In the next couple of weeks, my focus will be on&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;adding other high weak order solvers,&lt;/li&gt;
 &lt;li&gt;implementing adaptive time stepping.&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;More of our near-term goals are collected in this 
-&lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
+&lt;p&gt;More of our near-term goals are collected in this &lt;a href=&#34;https://github.com/SciML/StochasticDiffEq.jl/issues/182&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issue&lt;/a&gt;.&lt;/p&gt;
 &lt;h2 id=&#34;adjoint-sensitivity-methods-for-sdes&#34;&gt;Adjoint Sensitivity Methods for SDEs&lt;/h2&gt;
-&lt;p&gt;In 
-&lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
+&lt;p&gt;In &lt;a href=&#34;https://mitmath.github.io/18337/lecture10/estimation_identification&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;parameter estimation/inverse problems&lt;/a&gt;, one is interested to know the optimal choice of parameters &lt;code&gt;p&lt;/code&gt; such that a model &lt;code&gt;f(p)&lt;/code&gt;, e.g., a differential equation, optimally fits some data, y. The shooting method approaches this task by introducing some sort of loss function $L$. A common choice is the mean squared error&lt;/p&gt;
 &lt;p&gt;$$
 L = |f(p)-y|^2.
 $$&lt;/p&gt;
@@ -561,46 +496,43 @@ One possibility to obtain the gradient information for (stochastic) differential
 While forward mode AD is memory efficient, it scales poorly in time with increasing number of parameters. On the contrary, reverse-mode AD, i.e., a direct backpropagation through the solver, has a huge memory footprint.&lt;/p&gt;
 &lt;p&gt;Alternatively to the &amp;ldquo;direct&amp;rdquo; AD approaches, the &lt;strong&gt;adjoint sensitivity method&lt;/strong&gt; can be used&lt;sup id=&#34;fnref:3&#34;&gt;&lt;a href=&#34;#fn:3&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;3&lt;/a&gt;&lt;/sup&gt;. The adjoint sensitivity method is well known to compute gradients of solutions to ordinary differential equations (ODEs) with respect to the parameters and initial states entering the ODE. The method was recently generalized to SDEs&lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt;.
 Importantly, this new approach has different complexities in terms of memory consumption or computation time as compared with forward- or reverse-mode AD (NP vs N+P where N is the number of state variables and P is the number of parameters).&lt;/p&gt;
-&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
+&lt;p&gt;It turns out that the aforementioned gradients in the stochastic adjoint sensitivity method are given by solving an SDE with an &lt;strong&gt;augmented state backwards in time&lt;/strong&gt; launched at the end state of the forward evolution.  In other words, we first compute the forward time evolution of the model from the start time $t_0$ to the end time $t_1$. Subsequently, we reverse the SDE and run a second time evolution from $t_1$ to $t_0$. Please note that the authors in Ref. &lt;sup id=&#34;fnref1:4&#34;&gt;&lt;a href=&#34;#fn:4&#34; class=&#34;footnote-ref&#34; role=&#34;doc-noteref&#34;&gt;4&lt;/a&gt;&lt;/sup&gt; are implementing a slightly modfified version where the time evolution of the augmented state runs from $-t_1$ to $-t_0$. We however are indeed using the former variant as it allows us to reuse/generalize many functions that were implemented in the &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;DiffEqSensitivity&lt;/a&gt; package for ODE adjoints earlier.&lt;/p&gt;
 &lt;h3 id=&#34;reverse-sde-time-evolution&#34;&gt;Reverse SDE time evolution&lt;/h3&gt;
 &lt;p&gt;The reversion of an SDE is more difficult than the reversion of an ODE. However, for SDEs written in the Stratonovich sense, it turns out that reversion can be achieved by negative signs in front of the drift and diffusion terms.
 As one needs to follow the same trajectory backward, the noise sampled in the forward pass must be reconstructed.
 In general, we would like to use adaptive time-stepping solvers which require some form of interpolation for the noise values.
-After some fixes for the 
-&lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
+After some fixes for the &lt;a href=&#34;https://docs.sciml.ai/latest/features/noise_process/#Adaptive-NoiseWrapper-Example-1&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;available noise processes&lt;/a&gt;, we are now able to reverse a stochastic time evolution either by using &lt;code&gt;NoiseGrid&lt;/code&gt; which linearly interpolates between values of the noise on a given grid or by using a very general &lt;code&gt;NoiseWrapper&lt;/code&gt; which interpolates in a distributionally-exact manner based on Brownian bridges.&lt;/p&gt;
 &lt;p&gt;As an example, the code below computes first the forward evolution of an SDE&lt;/p&gt;
 &lt;p&gt;$$ dX  =  \alpha X dt + \beta X dW$$&lt;/p&gt;
 &lt;p&gt;with $\alpha=1.01$, $\beta=0.87$, $x(0)=1/2$, in the time span ($t_{0}=0$, $t_{1}=1)$. This forward evolution is shown in blue in the animation below. Subsequently, also the reverse time evolution (red) launched at time $t_{1}=1$ with initial value $x(t=1)$, propagated in negative time direction until $t_{0}=0$, is computed. We see that both trajectories match very well.&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;  using StochasticDiffEq, DiffEqNoiseProcess
-
-  α=1.01
-  β=0.87
-
-  dt = 1e-3
-  tspan = (0.0,1.0)
-  u₀=1/2
-
-  tarray =  collect(tspan[1]:dt:tspan[2])
-
-  f!(du,u,p,t) = du .= α*u
-  g!(du,u,p,t) = du .= β*u
-
-
-  prob = SDEProblem(f!,g!,[u₀],tspan)
-  sol =solve(prob,EulerHeun(),dt=dt,save_noise=true, adaptive=false)
-
-  _sol = deepcopy(sol) # to make sure the plot is correct
-  W1 = NoiseGrid(reverse!(_sol.t),reverse!(_sol.W.W))
-  prob1 = SDEProblem(f!,g!,sol[end],reverse(tspan),noise=W1)
-  sol1 = solve(prob1,EulerHeun(),dt=dt)
-&lt;/code&gt;&lt;/pre&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqNoiseProcess&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-3&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;  &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;α&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;du&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;du&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;.=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;β&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;deepcopy&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;c&#34;&gt;# to make sure the plot is correct&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;NoiseGrid&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;_sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;g!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;],&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;reverse&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tspan&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sol1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;
 
 
 
 
 
-  
 
 
 
@@ -612,18 +544,11 @@ After some fixes for the
 
 
 
-&lt;figure &gt;
 
-
-  &lt;a data-fancybox=&#34;&#34; href=&#34;https://frankschae.github.io/img/animation.gif&#34; &gt;
-
-
-  &lt;img src=&#34;https://frankschae.github.io/img/animation.gif&#34; alt=&#34;&#34;  &gt;
-&lt;/a&gt;
-
-
-
-&lt;/figure&gt;
+&lt;figure  &gt;
+  &lt;div class=&#34;d-flex justify-content-center&#34;&gt;
+    &lt;div class=&#34;w-100&#34; &gt;&lt;img src=&#34;animation.gif&#34; alt=&#34;&#34; loading=&#34;lazy&#34; data-zoomable /&gt;&lt;/div&gt;
+  &lt;/div&gt;&lt;/figure&gt;
 
 &lt;h3 id=&#34;gradients-of-diagonal-sdes&#34;&gt;Gradients of diagonal SDEs&lt;/h3&gt;
 &lt;p&gt;I have already started to implement the stochastic adjoint sensitivity method for SDEs possessing diagonal noise. Currently, only out-of-place SDE functions are supported but I am optimistic that soon also the inplace formulation works.&lt;/p&gt;
@@ -637,82 +562,80 @@ $$&lt;/p&gt;
 &lt;p&gt;acting on the solution $X(t)$ for some fixed time points $t_i$. Then, the analytical forms for the gradients here read&lt;/p&gt;
 &lt;p&gt;$$
 \begin{aligned}
-\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\&lt;br&gt;
+\frac{d L}{d \alpha} &amp;amp;= 2 \sum_i t_i |X(t_i)|^2 \\
 \frac{d L}{d \beta}  &amp;amp;= 2 \sum_i W(t_i) |X(t_i)|^2
 \end{aligned}
 $$&lt;/p&gt;
 &lt;p&gt;for $\alpha$ and $\beta$, respectively. We can confirm that this agrees with the gradients as obtained by the stochastic adjoint sensitivity method&lt;/p&gt;
-&lt;pre&gt;&lt;code class=&#34;language-julia&#34;&gt;using Test, LinearAlgebra
-using DiffEqSensitivity, StochasticDiffEq
-using Random
-
-seed = 100
-Random.seed!(seed)
-
-u₀ = [0.5]
-tstart = 0.0
-tend = 0.1
-dt = 0.005
-trange = (tstart, tend)
-t = tstart:dt:tend
-tarray = collect(t)
-
-function g(u,p,t)
-  sum(u.^2.0)
-end
-
-function dg!(out,u,p,t,i)
-  (out.=-2.0*u)
-end
-
-p2 = [1.01,0.87]
-
-f(u,p,t) = p[1]*u
-σ(u,p,t) = p[2]*u
-
-
-Random.seed!(seed)
-prob = SDEProblem(f,σ,u₀,trange,p2)
-sol = solve(prob,RKMil(interpretation=:Stratonovich),dt=tend/1e7,adaptive=false,save_noise=true)
-res_u0, res_p = adjoint_sensitivities(sol,EulerHeun(),dg!,t,dt=tend/1e7,sensealg=BacksolveAdjoint())
-
-
-noise = vec((@. sol.W(tarray)))
-Wextracted = [W[1][1] for W in noise]
-resp1 = 2*sum(@. tarray*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp2 = 2*sum(@. Wextracted*u₀^2*exp(2*(p2[1])*tarray+2*p2[2]*Wextracted))
-resp = [resp1, resp2]
-
-@test isapprox(res_p&#39;, resp, rtol = 1e-6)
-# True
-&lt;/code&gt;&lt;/pre&gt;
-&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
+&lt;div class=&#34;highlight&#34;&gt;&lt;pre tabindex=&#34;0&#34; class=&#34;chroma&#34;&gt;&lt;code class=&#34;language-julia&#34; data-lang=&#34;julia&#34;&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Test&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;LinearAlgebra&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;DiffEqSensitivity&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;StochasticDiffEq&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;using&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;100&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.5&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.0&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.1&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;0.005&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tstart&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;:&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:tend&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;collect&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;g&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.^&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;function&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;i&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;  &lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;out&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.=-&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;2.0&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;k&#34;&gt;end&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1.01&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;0.87&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;p&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Random&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;seed&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;SDEProblem&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;f&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;σ&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;trange&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;solve&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;prob&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;RKMil&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;interpretation&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;ss&#34;&gt;:Stratonovich&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;adaptive&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;false&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;save_noise&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;nb&#34;&gt;true&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;res_u0&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;adjoint_sensitivities&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;EulerHeun&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(),&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dg!&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;t&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;dt&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tend&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;/&lt;/span&gt;&lt;span class=&#34;mf&#34;&gt;1e7&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sensealg&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;=&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;BacksolveAdjoint&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;())&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;vec&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;((&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;sol&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;.&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;W&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;][&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;for&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;W&lt;/span&gt; &lt;span class=&#34;k&#34;&gt;in&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;noise&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;sum&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;nd&#34;&gt;@.&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;u₀&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;^&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;exp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;])&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;tarray&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;+&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;p2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;mi&#34;&gt;2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;*&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;Wextracted&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;))&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;p&#34;&gt;[&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;resp1&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp2&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;]&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;nd&#34;&gt;@test&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;isapprox&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;(&lt;/span&gt;&lt;span class=&#34;n&#34;&gt;res_p&lt;/span&gt;&lt;span class=&#34;o&#34;&gt;&amp;#39;&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;resp&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;,&lt;/span&gt; &lt;span class=&#34;n&#34;&gt;rtol&lt;/span&gt; &lt;span class=&#34;o&#34;&gt;=&lt;/span&gt; &lt;span class=&#34;mf&#34;&gt;1e-6&lt;/span&gt;&lt;span class=&#34;p&#34;&gt;)&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;span class=&#34;line&#34;&gt;&lt;span class=&#34;cl&#34;&gt;&lt;span class=&#34;c&#34;&gt;# True&lt;/span&gt;
+&lt;/span&gt;&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;p&gt;With respect to the adjoint sensitivity methods, we are planning to&lt;/p&gt;
 &lt;ul&gt;
 &lt;li&gt;finish the current backsolve adjoint version,&lt;/li&gt;
 &lt;li&gt;allow for computing the gradients of non-commuting SDEs,&lt;/li&gt;
 &lt;li&gt;implement also an interpolation adjoint version,&lt;/li&gt;
 &lt;li&gt;benchmark it with respect to AD approaches&lt;/li&gt;
 &lt;/ul&gt;
-&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open 
-&lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
+&lt;p&gt;in the upcoming weeks. For more information, the interested reader might take a look at the open &lt;a href=&#34;https://github.com/SciML/DiffEqSensitivity.jl/issues&#34; target=&#34;_blank&#34; rel=&#34;noopener&#34;&gt;issues&lt;/a&gt; in the DiffEqSensitivity package.&lt;/p&gt;
 &lt;p&gt;If you have any questions or comments, please don’t hesitate to contact me!&lt;/p&gt;
-&lt;section class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
+&lt;div class=&#34;footnotes&#34; role=&#34;doc-endnotes&#34;&gt;
 &lt;hr&gt;
 &lt;ol&gt;
-&lt;li id=&#34;fn:1&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009). &lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:1&#34;&gt;
+&lt;p&gt;Kristian Debrabant, Andreas Rößler, Applied Numerical Mathematics &lt;strong&gt;59&lt;/strong&gt;, 582–594 (2009).&amp;#160;&lt;a href=&#34;#fnref:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:1&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:2&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009). &lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:2&#34;&gt;
+&lt;p&gt;Andreas Rößler, Journal on Numerical Analysis &lt;strong&gt;47&lt;/strong&gt;, 1713–1738 (2009).&amp;#160;&lt;a href=&#34;#fnref:2&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:3&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012). &lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:3&#34;&gt;
+&lt;p&gt;Steven G. Johnson, &amp;ldquo;Notes on Adjoint Methods for 18.335.&amp;rdquo; Introduction to Numerical Methods (2012).&amp;#160;&lt;a href=&#34;#fnref:3&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
-&lt;li id=&#34;fn:4&#34; role=&#34;doc-endnote&#34;&gt;
-&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020). &lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
+&lt;li id=&#34;fn:4&#34;&gt;
+&lt;p&gt;Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud, arXiv preprint arXiv:2001.01328 (2020).&amp;#160;&lt;a href=&#34;#fnref:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&amp;#160;&lt;a href=&#34;#fnref1:4&#34; class=&#34;footnote-backref&#34; role=&#34;doc-backlink&#34;&gt;&amp;#x21a9;&amp;#xfe0e;&lt;/a&gt;&lt;/p&gt;
 &lt;/li&gt;
 &lt;/ol&gt;
-&lt;/section&gt;
+&lt;/div&gt;
 </description>
     </item>
     
diff --git a/tag/srk-methods/page/1/index.html b/tag/srk-methods/page/1/index.html
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-  <h1>stochastic optimal control</h1>
+  <h1>Stochastic Optimal Control</h1>
 
   
 
@@ -552,89 +758,272 @@ <h1>stochastic optimal control</h1>
 <div class="universal-wrapper">
   
 
+
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+
   
   
     
-    
-    
-    <div>
-      <h2><a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a></h2>
-      <div class="article-style">
-        
-          We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
-        
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+      <a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a>
+    </div>
 
     
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+        We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
+      </div>
+    </a>
     
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+  <span >
+      FS in Collaboration With Flemming Holtorf</span>, <span >
+      Julian Arnold</span>, <span >
+      Chris Rackauckas</span>, <span >
+      And Alan Edelman</span>
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diff --git a/tag/stochastic-optimal-control/index.xml b/tag/stochastic-optimal-control/index.xml
index 4ab2dad..036d8d2 100644
--- a/tag/stochastic-optimal-control/index.xml
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-      <title>stochastic optimal control</title>
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+      <title>Stochastic Optimal Control</title>
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index 242f4b5..3f07e16 100644
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+<!DOCTYPE html>
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+      
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     <section class="section-search-results">
 
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-      <h2><a href="/project/controlbounds/" >Performance Bounds for Quantum Control</a></h2>
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-          We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
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+        We compute bounds for the best attainable control performance that may serve as certificates of fundamental limitations or performance targets.
+      </div>
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+  <span >
+      FS in Collaboration With Flemming Holtorf</span>, <span >
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@@ -552,89 +758,327 @@ <h1>unsupervised learning</h1>
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-      <h2><a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a></h2>
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-          Data-driven methods based on sample instances of the state of a physical system as a function of the system's parameters.
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+
+
+
+
+  
+
+  
+  
+
+  
+
+
+
+
+
+
+  <script src="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.js" integrity="" crossorigin="anonymous"></script>
+
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+  
+  <script id="search-hit-fuse-template" type="text/x-template">
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@@ -642,42 +1086,18 @@ <h2><a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a></h
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+  
 
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-    
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-    
 
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-      
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-    
-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
-    
 
 
 
@@ -686,31 +1106,53 @@ <h2><a href="/project/ml_for_pt/" >Machine Learning for Phase Transitions</a></h
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
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-    
-    <span class="float-right" aria-hidden="true">
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-        </span>
-      </a>
-    </span>
-    
-  </p>
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-  </div>
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
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+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
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+  
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+
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 <div id="modal" class="modal fade" role="dialog">
@@ -723,7 +1165,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -738,5 +1181,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
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+
 </body>
 </html>
diff --git a/tag/unsupervised-learning/index.xml b/tag/unsupervised-learning/index.xml
index 328ba2c..d69c91d 100644
--- a/tag/unsupervised-learning/index.xml
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+    <title>Unsupervised Learning | FS</title>
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+    <description>Unsupervised Learning</description>
+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Tue, 09 Mar 2021 23:56:43 +0100</lastBuildDate>
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-      <title>unsupervised learning</title>
+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
+      <title>Unsupervised Learning</title>
       <link>https://frankschae.github.io/tag/unsupervised-learning/</link>
     </image>
     
diff --git a/tag/unsupervised-learning/page/1/index.html b/tag/unsupervised-learning/page/1/index.html
index 6f1ae24..9891eb5 100644
--- a/tag/unsupervised-learning/page/1/index.html
+++ b/tag/unsupervised-learning/page/1/index.html
@@ -1 +1,10 @@
-<!DOCTYPE html><html><head><title>https://frankschae.github.io/tag/unsupervised-learning/</title><link rel="canonical" href="https://frankschae.github.io/tag/unsupervised-learning/"/><meta name="robots" content="noindex"><meta charset="utf-8" /><meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/unsupervised-learning/" /></head></html>
\ No newline at end of file
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/tag/unsupervised-learning/</title>
+    <link rel="canonical" href="https://frankschae.github.io/tag/unsupervised-learning/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tag/unsupervised-learning/">
+  </head>
+</html>
diff --git a/tags/index.html b/tags/index.html
index ced4a34..55d8023 100644
--- a/tags/index.html
+++ b/tags/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
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-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
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-  
 
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+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
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-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tags/">
+
+
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+
+
+
 
   
 
+
+
+
+
+
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+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
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-  <meta name="theme-color" content="#2962ff">
+    
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+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
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+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
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+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
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+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
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+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
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+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
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+
 
 
 
 
-  
 
 
   
   
-
   
-  <link rel="alternate" href="/tags/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
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+<link rel="canonical" href="https://frankschae.github.io/tags/" />
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+
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+  <link rel="manifest" href="/manifest.webmanifest" />
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+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
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+
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+  <meta property="twitter:creator" content="@GetResearchDev" />
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+<meta property="og:type" content="website" />
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-    <meta property="og:updated_time" content="2023-04-29T17:26:49-04:00">
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+  <title>Tags | FS</title>
 
+  
+  
+  
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-  
 
 
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-  <title>Tags | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
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         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
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       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
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@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
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-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
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-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
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+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
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+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
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+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
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+        <a class="navbar-brand" href="/">FS</a>
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+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
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+          
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-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
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-        
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-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
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-        'project': "Projects",
-        'publication' : "Publications",
-        'talk' : "Talks"
-        };
-    </script>
-    
 
-    
-    
 
-    
-    
-    <script id="search-hit-fuse-template" type="text/x-template">
-      <div class="search-hit" id="summary-{{key}}">
+
+
+  
+  <script id="search-hit-fuse-template" type="text/x-template">
+    <div class="search-hit" id="summary-{{key}}">
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@@ -739,42 +1652,18 @@ <h2><a href="/tag/event-handling/" >Event handling</a></h2>
           <p class="search-hit-description">{{snippet}}</p>
         </div>
       </div>
-      </div>
-    </script>
-    
+    </div>
+  </script>
+  
+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
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+  
 
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-    <script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/3.2.1/fuse.min.js" integrity="sha256-VzgmKYmhsGNNN4Ph1kMW+BjoYJM2jV5i4IlFoeZA9XI=" crossorigin="anonymous"></script>
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-    
-    
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-    
-      
-    
-    
-    
-    
-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
-    
 
 
 
@@ -783,32 +1672,54 @@ <h2><a href="/tag/event-handling/" >Event handling</a></h2>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
-    <a href="https://gohugo.io" target="_blank" rel="noopener">Hugo</a>.
 
-    
-    <span class="float-right" aria-hidden="true">
-      <a href="#" class="back-to-top">
-        <span class="button_icon">
-          <i class="fas fa-chevron-up fa-2x"></i>
-        </span>
-      </a>
-    </span>
-    
-  </p>
-</footer>
 
-  </div>
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
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+
+
+
+  
   
 
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
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+
   
 <div id="modal" class="modal fade" role="dialog">
   <div class="modal-dialog">
@@ -820,7 +1731,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -835,5 +1747,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
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+
+
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+
 </body>
 </html>
diff --git a/tags/index.xml b/tags/index.xml
index 829dd70..85a73ea 100644
--- a/tags/index.xml
+++ b/tags/index.xml
@@ -5,9 +5,9 @@
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       <atom:link href="https://frankschae.github.io/tags/index.xml" rel="self" type="application/rss+xml" />
     <description>Tags</description>
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+    <generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Sat, 29 Apr 2023 17:26:49 -0400</lastBuildDate>
     <image>
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+      <url>https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png</url>
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       <link>https://frankschae.github.io/tags/</link>
     </image>
diff --git a/tags/page/1/index.html b/tags/page/1/index.html
index fbed564..b6bee51 100644
--- a/tags/page/1/index.html
+++ b/tags/page/1/index.html
@@ -1 +1,10 @@
-<!DOCTYPE html><html><head><title>https://frankschae.github.io/tags/</title><link rel="canonical" href="https://frankschae.github.io/tags/"/><meta name="robots" content="noindex"><meta charset="utf-8" /><meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tags/" /></head></html>
\ No newline at end of file
+<!DOCTYPE html>
+<html lang="en-us">
+  <head>
+    <title>https://frankschae.github.io/tags/</title>
+    <link rel="canonical" href="https://frankschae.github.io/tags/">
+    <meta name="robots" content="noindex">
+    <meta charset="utf-8">
+    <meta http-equiv="refresh" content="0; url=https://frankschae.github.io/tags/">
+  </head>
+</html>
diff --git a/tags/page/2/index.html b/tags/page/2/index.html
index 0d9d263..407a22f 100644
--- a/tags/page/2/index.html
+++ b/tags/page/2/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tags/">
+
+
+
+
+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
-  <script src="/js/mathjax-config.js"></script>
   
 
   
+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
+
   
   
   
     
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+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1.9.4/css/academicons.min.css" integrity="sha512-IW0nhlW5MgNydsXJO40En2EoCkTTjZhI3yuODrZIc8cQ4h1XcF53PsqDHa09NqnkXuIe0Oiyyj171BqZFwISBw==" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
+
     
     
-      
     
     
       
       
-        
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/github.min.css" crossorigin="anonymous" title="hl-light">
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-
     
     
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+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +141,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
+  
+  
+  
+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
+
+  
+  
   
 
   
   
   
   
-  <link rel="stylesheet" href="/css/academic.css">
+  
+  
+  
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+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
+  
 
   
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+
+
+
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+
+
+
+
+
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+
+
+
+
+
+
+
+
+
+
+
 
 
 
 
-  
 
 
   
   
-
   
-  <link rel="alternate" href="/tags/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
+
+
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+
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+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
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+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tags/" />
+<link rel="canonical" href="https://frankschae.github.io/tags/" />
+
+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
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+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
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+
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+
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+
 
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+
+
+
+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
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+
+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/tags/" />
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+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
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+
   
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+    <link rel="alternate" href="/tags/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2023-04-29T17:26:49-04:00">
+
   
 
+
   
+  <title>Tags | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>Tags | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
+
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+  
+  
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+  
+  
+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
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   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
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         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
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@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
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-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
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-            data-target="#navbar-content" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
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+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
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+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
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+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
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+      </button>
+      
+
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+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
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+      
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+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
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+        <ul class="navbar-nav d-md-inline-flex">
           
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-        
-        
-        
-        
-        
-        
           
+
           
           
-            
           
+            
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
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+          
+          
+          
+          
 
-        
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+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
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-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
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+
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-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
-        </li>
+          
+          
+          
+          
 
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+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
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+                
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+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
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+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
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+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
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+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
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+      <a href="/tag/differentiable-programming/" >Differentiable Programming</a>
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-    <script id="search-hit-fuse-template" type="text/x-template">
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+
+
+  
+  <script id="search-hit-fuse-template" type="text/x-template">
+    <div class="search-hit" id="summary-{{key}}">
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@@ -741,42 +1654,18 @@ <h2><a href="/tag/machine-learning/" >Machine learning</a></h2>
           <p class="search-hit-description">{{snippet}}</p>
         </div>
       </div>
-      </div>
-    </script>
-    
+    </div>
+  </script>
+  
+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
+    <script src="https://cdn.jsdelivr.net/gh/julmot/mark.js@8.11.1/dist/jquery.mark.min.js" integrity="sha512-mhbv5DqBMgrWL+32MmsDOt/OAvqr/cHimk6B8y/bx/xS88MVkYGPiVv2ixKVrkywF2qHplNRUvFsAHUdxZ3Krg==" crossorigin="anonymous"></script>
+  
 
-    
-    
-    <script src="https://cdnjs.cloudflare.com/ajax/libs/fuse.js/3.2.1/fuse.min.js" integrity="sha256-VzgmKYmhsGNNN4Ph1kMW+BjoYJM2jV5i4IlFoeZA9XI=" crossorigin="anonymous"></script>
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-    
 
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-    
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-    
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-    
-    
-    
-    
-    
-    
-    
-    
-    
-      
-    
-    
-    
-    
-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
-    
 
 
 
@@ -785,32 +1674,54 @@ <h2><a href="/tag/machine-learning/" >Machine learning</a></h2>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
-    <a href="https://gohugo.io" target="_blank" rel="noopener">Hugo</a>.
 
-    
-    <span class="float-right" aria-hidden="true">
-      <a href="#" class="back-to-top">
-        <span class="button_icon">
-          <i class="fas fa-chevron-up fa-2x"></i>
-        </span>
-      </a>
-    </span>
-    
-  </p>
-</footer>
 
-  </div>
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+  
   
 
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
+
   
 <div id="modal" class="modal fade" role="dialog">
   <div class="modal-dialog">
@@ -822,7 +1733,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -837,5 +1749,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
+
+
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+
+
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+
+
 </body>
 </html>
diff --git a/tags/page/3/index.html b/tags/page/3/index.html
index ba422e1..509f099 100644
--- a/tags/page/3/index.html
+++ b/tags/page/3/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
+
+
+
+
+
+
 
   
   
-  
-    
-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tags/">
+
+
+
+
+
+
+
+
 
   
 
+
+
+
+
+
+
+  
+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
-  <script src="/js/mathjax-config.js"></script>
   
 
   
+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
+
   
   
   
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/academicons/1.8.6/css/academicons.min.css" integrity="sha256-uFVgMKfistnJAfoCUQigIl+JfUaP47GrRKjf6CTPVmw=" crossorigin="anonymous">
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-
     
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1.9.4/css/academicons.min.css" integrity="sha512-IW0nhlW5MgNydsXJO40En2EoCkTTjZhI3yuODrZIc8cQ4h1XcF53PsqDHa09NqnkXuIe0Oiyyj171BqZFwISBw==" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
+
     
     
-      
     
     
       
       
-        
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/github.min.css" crossorigin="anonymous" title="hl-light">
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/dracula.min.css" crossorigin="anonymous" title="hl-dark" disabled>
-        
-      
     
-
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.css" integrity="sha256-SHMGCYmST46SoyGgo4YR/9AlK1vf3ff84Aq9yK4hdqM=" crossorigin="anonymous">
     
 
     
-
     
     
-      
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +141,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
+  
+  
+  
+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
+
+  
+  
   
 
   
   
   
   
-  <link rel="stylesheet" href="/css/academic.css">
+  
+  
+  
+    
+    
+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
+  
 
   
+  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
 
 
 
 
-  
 
 
   
   
-
   
-  <link rel="alternate" href="/tags/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
+
+
+
+
+
   
 
-  <link rel="manifest" href="/index.webmanifest">
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+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
+
+
+
+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tags/" />
+<link rel="canonical" href="https://frankschae.github.io/tags/" />
+
+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
+
+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
+<link rel="apple-touch-icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_180x180_fill_lanczos_center_3.png" />
+
+<meta name="theme-color" content="#1565c0" />
+
+
+
+
+
+
+
+
 
-  <link rel="canonical" href="https://frankschae.github.io/tags/">
 
   
   
+
+
+
+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
+<meta property="twitter:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" />
+
+
+
   
+
+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/tags/" />
+<meta property="og:title" content="Tags | FS" />
+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
+
   
+    <meta property="og:updated_time" content="2023-04-29T17:26:49-04:00" />
   
+
+
+
+
+
+
+
+
+
+
   
   
-    
-    
   
+
   
-  <meta property="twitter:card" content="summary">
   
-  <meta property="og:site_name" content="FS">
-  <meta property="og:url" content="https://frankschae.github.io/tags/">
-  <meta property="og:title" content="Tags | FS">
-  <meta property="og:description" content="Postdoctoral researcher"><meta property="og:image" content="https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png">
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+    <link rel="alternate" href="/tags/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2023-04-29T17:26:49-04:00">
+
   
 
+
   
+  <title>Tags | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>Tags | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
+
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
+
+  
+
+
+<aside class="search-modal" id="search">
   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
           <h1>Search</h1>
         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
+  
   
 
 
@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    <button type="button" class="navbar-toggler" data-toggle="collapse"
-            data-target="#navbar-content" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
-    <span><i class="fas fa-bars"></i></span>
-    </button>
-    
 
-    
-    <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    
-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
-        
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
 
-        
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
-        
-        
+        <ul class="navbar-nav d-md-inline-flex">
           
-        
 
-        
-        
-        
-        
-        
-        
           
+
           
           
-            
           
+            
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
-        
-        
-        
           
-        
+          
 
-        
-        
-        
-        
-        
-        
           
+
           
           
           
-        
+            
+          
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#software"><span>Open Source Software</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
+          
+          
+
+          
+
+          
+          
+          
+            
+          
+
+          
+
+          
+          
+          
+          
+
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
 
         
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
@@ -461,63 +681,852 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
+
         
         
-          
-          
-          
-            
-          
-          
         
-
-        <li class="nav-item">
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-    <script src="/js/academic.min.37431be2d92d7fb0160054761ab79602.js"></script>
 
-    
 
 
 
@@ -785,32 +1674,54 @@ <h2><a href="/tag/bosonic-systems/" >Bosonic systems</a></h2>
 
   
   
-  <div class="container">
-    <footer class="site-footer">
+  
+  
+  
+  
   
 
-  <p class="powered-by">
-    © 2020 Frank Schäfer &middot; 
 
-    Powered by the
-    <a href="https://sourcethemes.com/academic/" target="_blank" rel="noopener">Academic theme</a> for
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-    
-    <span class="float-right" aria-hidden="true">
-      <a href="#" class="back-to-top">
-        <span class="button_icon">
-          <i class="fas fa-chevron-up fa-2x"></i>
-        </span>
-      </a>
-    </span>
-    
-  </p>
-</footer>
 
-  </div>
+
+
+
+
+
+
+
+
+
+
+
+
+
+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
+
+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
+
+
+
+
+
+
+
+
+  
   
 
+
+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
+
+
+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
+
   
 <div id="modal" class="modal fade" role="dialog">
   <div class="modal-dialog">
@@ -822,7 +1733,8 @@ <h5 class="modal-title">Cite</h5>
         </button>
       </div>
       <div class="modal-body">
-        <pre><code class="tex hljs"></code></pre>
+        
+        <pre><code></code></pre>
       </div>
       <div class="modal-footer">
         <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
@@ -837,5 +1749,25 @@ <h5 class="modal-title">Cite</h5>
   </div>
 </div>
 
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
+
+
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 </html>
diff --git a/tags/page/4/index.html b/tags/page/4/index.html
index c2de868..a9625b5 100644
--- a/tags/page/4/index.html
+++ b/tags/page/4/index.html
@@ -1,91 +1,106 @@
-<!DOCTYPE html><html lang="en-us" >
+<!DOCTYPE html>
+<!-- This site was created with Hugo Blox. https://hugoblox.com -->
+<!-- Last Published: April 14, 2024 --><html lang="en-us" >
 
-<head>
-
-  <meta charset="utf-8">
-  <meta name="viewport" content="width=device-width, initial-scale=1">
-  <meta http-equiv="X-UA-Compatible" content="IE=edge">
-  <meta name="generator" content="Source Themes Academic 4.8.0">
-
-  
 
+<head>
+  <meta charset="utf-8" />
+  <meta name="viewport" content="width=device-width, initial-scale=1" />
+  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
   
   
   
+    <meta name="generator" content="Hugo Blox Builder 5.9.7" />
   
-  
-    
-    
-    
+
   
   
 
-  <meta name="author" content="Frank Schäfer">
+
+
+
+
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+
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+
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+
 
   
   
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-  
-  <meta name="description" content="Postdoctoral researcher">
 
-  
-  <link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tags/">
+
+
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+
+
 
   
 
+
+
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+  
+  <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin />
+  
 
   
   
   
-  <meta name="theme-color" content="#2962ff">
+    
+      
+      <link rel="preload" as="style" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap">
+      <link rel="stylesheet" href="https://fonts.googleapis.com/css2?family=Montserrat:wght@400;700&family=Roboto+Mono&family=Roboto:wght@400;700&display=swap" media="print" onload="this.media='all'">
+    
   
 
   
   
+
   
-  <script src="/js/mathjax-config.js"></script>
   
 
   
+
+  <link rel="stylesheet" href="/css/vendor-bundle.min.26c458e6907dc03073573976b7f4044e.css" media="print" onload="this.media='all'">
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+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/gh/jpswalsh/academicons@1.9.4/css/academicons.min.css" integrity="sha512-IW0nhlW5MgNydsXJO40En2EoCkTTjZhI3yuODrZIc8cQ4h1XcF53PsqDHa09NqnkXuIe0Oiyyj171BqZFwISBw==" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
+
     
     
-      
     
     
       
       
-        
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/github.min.css" crossorigin="anonymous" title="hl-light">
-          <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/9.18.1/styles/dracula.min.css" crossorigin="anonymous" title="hl-dark" disabled>
-        
-      
     
-
     
-    <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/leaflet/1.5.1/leaflet.css" integrity="sha256-SHMGCYmST46SoyGgo4YR/9AlK1vf3ff84Aq9yK4hdqM=" crossorigin="anonymous">
     
 
     
-
     
     
-      
+      <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/leaflet@1.7.1/dist/leaflet.min.css" integrity="" crossorigin="anonymous" media="print" onload="this.media='all'">
+    
 
-      
-      
+    
 
-      
     
+    
+      
       
 
       
@@ -94,6 +109,7 @@
       
     
       
+      
 
       
       
@@ -101,6 +117,7 @@
       
     
       
+      
 
       
       
@@ -108,6 +125,7 @@
       
     
       
+      
 
       
       
@@ -115,6 +133,7 @@
       
     
       
+      
 
       
       
@@ -122,6 +141,7 @@
       
     
       
+      
 
       
       
@@ -129,6 +149,7 @@
       
     
       
+      
 
       
       
@@ -136,6 +157,7 @@
       
     
       
+      
 
       
       
@@ -143,6 +165,7 @@
       
     
       
+      
 
       
       
@@ -150,37 +173,32 @@
       
     
       
+      
 
       
       
 
       
-        <script src="https://cdnjs.cloudflare.com/ajax/libs/lazysizes/5.1.2/lazysizes.min.js" integrity="sha256-Md1qLToewPeKjfAHU1zyPwOutccPAm5tahnaw7Osw0A=" crossorigin="anonymous" async></script>
-      
     
       
-
-      
       
 
       
-    
-      
-
       
+        
       
 
       
     
       
+      
 
       
       
 
-      
-        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js" integrity="" crossorigin="anonymous" async></script>
       
     
+      
       
 
       
@@ -188,85 +206,193 @@
 
       
     
-
   
 
   
   
   
-  <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Montserrat:400,700%7CRoboto:400,400italic,700%7CRoboto+Mono&display=swap">
+  
+  
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+  <link rel="stylesheet" href="/css/wowchemy.f6689966c0a10712f95f034011917db0.css" />
+
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+  
   
 
   
   
   
   
-  <link rel="stylesheet" href="/css/academic.css">
+  
+  
+  
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+    <link rel="stylesheet" href="/css/libs/chroma/github-light.min.css" title="hl-light" media="print" onload="this.media='all'" >
+    <link rel="stylesheet" href="/css/libs/chroma/dracula.min.css" title="hl-dark" media="print" onload="this.media='all'" disabled>
+  
 
   
+  
+
+
+
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+
+
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+
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+
+
+
+
+
+
+
+
+
+
+
 
 
 
 
-  
 
 
   
   
-
   
-  <link rel="alternate" href="/tags/index.xml" type="application/rss+xml" title="FS">
+
+
+
+
+
+
+  <meta name="author" content="Frank Schäfer" />
+
+
+
+
+
   
 
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+<meta name="description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." />
+
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+
+<link rel="alternate" hreflang="en-us" href="https://frankschae.github.io/tags/" />
+<link rel="canonical" href="https://frankschae.github.io/tags/" />
+
+
+
+  <link rel="manifest" href="/manifest.webmanifest" />
+
+
+
+<link rel="icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_32x32_fill_lanczos_center_3.png" />
+<link rel="apple-touch-icon" type="image/png" href="/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_180x180_fill_lanczos_center_3.png" />
+
+<meta name="theme-color" content="#1565c0" />
+
+
+
+
+
+
+
+
 
-  <link rel="canonical" href="https://frankschae.github.io/tags/">
 
   
   
+
+
+
+
+
+
+<meta property="twitter:card" content="summary" />
+
+  <meta property="twitter:site" content="@GetResearchDev" />
+  <meta property="twitter:creator" content="@GetResearchDev" />
+<meta property="twitter:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" />
+
+
+
   
+
+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/tags/" />
+<meta property="og:title" content="Tags | FS" />
+<meta property="og:description" content="A highly-customizable Hugo academic resume theme powered by Hugo Blox Builder." /><meta property="og:image" content="https://frankschae.github.io/media/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_3.png" /><meta property="og:locale" content="en-us" />
+
   
+    <meta property="og:updated_time" content="2023-04-29T17:26:49-04:00" />
   
+
+
+
+
+
+
+
+
+
+
   
   
-    
-    
   
+
   
-  <meta property="twitter:card" content="summary">
   
-  <meta property="og:site_name" content="FS">
-  <meta property="og:url" content="https://frankschae.github.io/tags/">
-  <meta property="og:title" content="Tags | FS">
-  <meta property="og:description" content="Postdoctoral researcher"><meta property="og:image" content="https://frankschae.github.io/images/icon_hu0b7a4cb9992c9ac0e91bd28ffd38dd00_9727_512x512_fill_lanczos_center_2.png">
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+    <link rel="alternate" href="/tags/index.xml" type="application/rss+xml" title="FS" />
   
-    <meta property="og:updated_time" content="2023-04-29T17:26:49-04:00">
+
   
 
+
   
+  <title>Tags | FS</title>
 
+  
+  
+  
+  
 
 
 
-  
 
 
-  
 
 
 
 
 
-  <title>Tags | FS</title>
 
 </head>
 
-<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" >
 
-  <aside class="search-results" id="search">
+<body id="top" data-spy="scroll" data-offset="70" data-target="#TableOfContents" class="page-wrapper   "  >
+
+  
+  
+  
+  
+  
+  
+  
+  
+  
+  <script src="/js/wowchemy-init.min.9e4214442a7711d35691acd58f6f6361.js"></script>
+
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+
+
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   <div class="container">
     <section class="search-header">
 
@@ -275,17 +401,23 @@
           <h1>Search</h1>
         </div>
         <div class="col-6 col-search-close">
-          <a class="js-search" href="#"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
         </div>
       </div>
 
       <div id="search-box">
         
         <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
-        autocomplete="off" autocorrect="off" spellcheck="false" type="search">
+        autocomplete="off" autocorrect="off" spellcheck="false" type="search" class="form-control"
+        aria-label="Search...">
         
       </div>
 
+      
+      
+
+      
+
     </section>
     <section class="section-search-results">
 
@@ -298,6 +430,12 @@ <h1>Search</h1>
 </aside>
 
 
+
+  <div class="page-header header--fixed">
+  
+  
+  
+  
   
 
 
@@ -306,153 +444,235 @@ <h1>Search</h1>
 
 
 
-<nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
-  <div class="container">
 
-    
-    <div class="d-none d-lg-inline-flex">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
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-    </button>
-    
 
-    
-    <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
-      <a class="navbar-brand" href="/">FS</a>
-    </div>
-    
 
-    
-    
-    <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
+
+<header>
+  <nav class="navbar navbar-expand-lg navbar-light compensate-for-scrollbar" id="navbar-main">
+    <div class="container-xl">
 
       
-      <ul class="navbar-nav d-md-inline-flex">
-        
+      <div class="d-none d-lg-inline-flex">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
 
-        
+      
+      <button type="button" class="navbar-toggler" data-toggle="collapse"
+              data-target="#navbar-content" aria-controls="navbar-content" aria-expanded="false" aria-label="Toggle navigation">
+      <span><i class="fas fa-bars"></i></span>
+      </button>
+      
+
+      
+      <div class="navbar-brand-mobile-wrapper d-inline-flex d-lg-none">
+        <a class="navbar-brand" href="/">FS</a>
+      </div>
+      
+
+      
+      
+      <div class="navbar-collapse main-menu-item collapse justify-content-start" id="navbar-content">
 
         
-        
-        
+        <ul class="navbar-nav d-md-inline-flex">
           
-        
 
-        
-        
-        
-        
-        
-        
           
+
           
           
-            
           
+            
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#about"><span>Home</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#about"><span>Home</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#posts"><span>Posts</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#posts"><span>Posts</span></a>
+          </li>
+
+          
+          
 
-        
-        
-        
           
-        
 
-        
-        
-        
-        
-        
-        
           
           
           
             
           
+
           
-        
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#projects"><span>Research projects</span></a>
-        </li>
+          
+          
+          
+          
 
-        
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
 
-        
+          <li class="nav-item">
+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
+          </li>
 
-        
-        
-        
           
-        
+          
 
-        
-        
-        
-        
-        
-        
           
+
           
           
           
-        
+            
+          
 
-        <li class="nav-item">
-          <a class="nav-link " href="/#software"><span>Open Source Software</span></a>
-        </li>
+          
 
-        
-        
+          
+          
+          
+          
 
-        
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
+          </li>
+
+          
+          
+
+          
+
+          
+          
+          
+            
+          
+
+          
+
+          
+          
+          
+          
+
+          
+            
+              
+              
+            
+            
+              
+              
+              
+                
+              
+              
+            
+          
+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
 
         
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
         
         
           
@@ -461,65 +681,51 @@ <h1>Search</h1>
         
         
         
+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
         
+
         
         
-          
-          
-          
-            
-          
-          
         
-
-        <li class="nav-item">
-          <a class="nav-link " href="/#contact"><span>Contact</span></a>
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
+            </a>
+            <a href="#" class="dropdown-item js-set-theme-auto">
+              <span>Automatic</span>
+            </a>
+          </div>
         </li>
+        
 
         
         
 
-      
-
-        
       </ul>
-    </div>
-
-    <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
-      
-      <li class="nav-item">
-        <a class="nav-link js-search" href="#"><i class="fas fa-search" aria-hidden="true"></i></a>
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-      
-
-      
-      <li class="nav-item dropdown theme-dropdown">
-        <a href="#" class="nav-link js-theme-selector" data-toggle="dropdown" aria-haspopup="true">
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-        </a>
-        <div class="dropdown-menu">
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-            <span>Light</span>
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-          <a href="#" class="dropdown-item js-set-theme-auto">
-            <span>Automatic</span>
-          </a>
-        </div>
-      </li>
-      
 
-      
+    </div>
+  </nav>
+</header>
 
-    </ul>
 
   </div>
-</nav>
 
+  <div class="page-body">
+    
+    
+    
+
+    
 
-  
 
 
 
@@ -552,169 +758,411 @@ <h1>Tags</h1>
 <div class="universal-wrapper">
   
 
+
   
+
   
-    
-    
-    
-    <div>
-      <h2><a href="/tag/multiple-scattering/" >multiple scattering</a></h2>
-      <div class="article-style">
-        
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+<meta property="og:type" content="website" />
+<meta property="og:site_name" content="FS" />
+<meta property="og:url" content="https://frankschae.github.io/talk/example-talk/" />
+<meta property="og:title" content="Example Talk | FS" />
+<meta property="og:description" content="An example talk using Hugo Blox Builder&#39;s Markdown slides feature." /><meta property="og:image" content="https://frankschae.github.io/talk/example-talk/featured.jpg" /><meta property="og:locale" content="en-us" />
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+      property="article:published_time"
+      content="2017-01-01T00:00:00&#43;00:00"
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+  },
+  "name": "Example Talk",
+  
+  "location": {
+    "@type": "Place",
+    "name": "Hugo Blox Builder HQ",
+    "address": {
+      "@type": "PostalAddress",
+      "streetAddress": "450 Serra Mall",
+      "addressLocality": "Stanford",
+      "postalCode": "94305",
+      "addressRegion": "CA",
+      "addressCountry": "United States"
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+  "image": [
+    "https://frankschae.github.io/talk/example-talk/featured.jpg"
+  ],
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+  "startDate": "2030-06-01T13:00:00Z",
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+  "endDate": "2030-06-01T15:00:00Z",
+  
+  
+  "performer": {
+    "@type": "Person",
+    "name": "Frank Schäfer"
+  },
+  
+  "description": "An example talk using Hugo Blox Builder's Markdown slides feature."
+}
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+  <title>Example Talk | FS</title>
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+      <div class="row no-gutters justify-content-between mb-3">
+        <div class="col-6">
+          <h1>Search</h1>
+        </div>
+        <div class="col-6 col-search-close">
+          <a class="js-search" href="#" aria-label="Close"><i class="fas fa-times-circle text-muted" aria-hidden="true"></i></a>
+        </div>
+      </div>
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+      <div id="search-box">
+        
+        <input name="q" id="search-query" placeholder="Search..." autocapitalize="off"
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+        <a class="navbar-brand" href="/">FS</a>
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+            <a class="nav-link " href="/#about"><span>Home</span></a>
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+            <a class="nav-link " href="/#projects"><span>Research projects</span></a>
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+            <a class="nav-link " href="/#software"><span>Open source software</span></a>
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+              
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+                
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+              
+            
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+
+          <li class="nav-item">
+            <a class="nav-link " href="/#contact"><span>Contact</span></a>
+          </li>
+
+          
+          
+
+        
+
+          
+        </ul>
+      </div>
+
+      <ul class="nav-icons navbar-nav flex-row ml-auto d-flex pl-md-2">
+
+        
+        
+          
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+        <li class="nav-item">
+          <a class="nav-link js-search" href="#" aria-label="Search"><i class="fas fa-search" aria-hidden="true"></i></a>
+        </li>
+        
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+        
+        
+        
+        <li class="nav-item dropdown theme-dropdown">
+          <a href="#" class="nav-link" data-toggle="dropdown" aria-haspopup="true" aria-label="Display preferences">
+            <i class="fas fa-moon" aria-hidden="true"></i>
+          </a>
+          <div class="dropdown-menu">
+            <a href="#" class="dropdown-item js-set-theme-light">
+              <span>Light</span>
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+            <a href="#" class="dropdown-item js-set-theme-dark">
+              <span>Dark</span>
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+              <span>Automatic</span>
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+  
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+<div class="article-container pt-3">
+  <h1>Example Talk</h1>
+
+  
+
+  
+
+
+<div class="article-metadata">
+
+  
+  
+
+  
+
+  
+
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+  
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+  <a class="btn btn-outline-primary btn-page-header" href="/slides/example/" target="_blank">
+    Slides
+  </a>
+  
+
+
+
+
+
+  
+  
+  
+    
+  
+  
+  
+  
+  
+    
+  
+  <a class="btn btn-outline-primary btn-page-header" href="https://twitter.com/georgecushen" target="_blank" rel="noopener">
+    <i class="fab fa-twitter mr-1"></i>Follow</a>
+
+
+</div>
+
+
+</div>
+
+
+<div class="article-header article-container featured-image-wrapper mt-4 mb-4" style="max-width: 720px; max-height: 514px;">
+  <div style="position: relative">
+    <img src="/talk/example-talk/featured_hu3d03a01dcc18bc5be0e67db3d8d209a6_620088_bf168a4e69f6a44b8c924d736876253f.webp" width="720" height="514" alt="" class="featured-image">
+    <span class="article-header-caption">Image credit: <a href="https://unsplash.com/photos/bzdhc5b3Bxs" target="_blank" rel="noopener"><strong>Unsplash</strong></a></span>
+  </div>
+</div>
+
+
+
+  <div class="article-container">
+
+    
+      <h3>Abstract</h3>
+      <p class="pub-abstract">Lorem ipsum dolor sit amet, consectetur adipiscing elit. Duis posuere tellusac convallis placerat. Proin tincidunt magna sed ex sollicitudin condimentum. Sed ac faucibus dolor, scelerisque sollicitudin nisi. Cras purus urna, suscipit quis sapien eu, pulvinar tempor diam.</p>
+    
+
+    <div class="row">
+      <div class="col-md-1"></div>
+      <div class="col-md-10">
+        <div class="row">
+          <div class="col-12 col-md-3 pub-row-heading">Date</div>
+          <div class="col-12 col-md-9">
+            Jun 1, 2030 1:00 PM &mdash; 3:00 PM
+          </div>
+        </div>
+      </div>
+      <div class="col-md-1"></div>
+    </div>
+    <div class="d-md-none space-below"></div>
+
+    
+    <div class="row">
+      <div class="col-md-1"></div>
+      <div class="col-md-10">
+        <div class="row">
+          <div class="col-12 col-md-3 pub-row-heading">Event</div>
+          <div class="col-12 col-md-9">
+            <a href="https://example.org" target="_blank" rel="noopener">
+            Hugo Blox Builder Conference
+            </a>
+          </div>
+        </div>
+      </div>
+      <div class="col-md-1"></div>
+    </div>
+    <div class="d-md-none space-below"></div>
+    
+
+    
+    <div class="row">
+      <div class="col-md-1"></div>
+      <div class="col-md-10">
+        <div class="row">
+          <div class="col-12 col-md-3 pub-row-heading">Location</div>
+          <div class="col-12 col-md-9">Hugo Blox Builder HQ</div>
+          
+            <div class="col-md-3"></div>
+            <div class="col-12 col-md-9">450 Serra Mall, Stanford, CA 94305</div>
+          
+        </div>
+      </div>
+      <div class="col-md-1"></div>
+    </div>
+    <div class="d-md-none space-below"></div>
+    
+
+    <div class="space-below"></div>
+
+    <div class="article-style">
+      <div class="alert alert-note">
+  <div>
+    Click on the <strong>Slides</strong> button above to view the built-in slides feature.
+  </div>
+</div>
+<p>Slides can be added in a few ways:</p>
+<ul>
+<li><strong>Create</strong> slides using Hugo Blox Builder&rsquo;s <a href="https://docs.hugoblox.com/reference/content-types/" target="_blank" rel="noopener"><em>Slides</em></a> feature and link using <code>slides</code> parameter in the front matter of the talk file</li>
+<li><strong>Upload</strong> an existing slide deck to <code>static/</code> and link using <code>url_slides</code> parameter in the front matter of the talk file</li>
+<li><strong>Embed</strong> your slides (e.g. Google Slides) or presentation video on this page using <a href="https://docs.hugoblox.com/reference/markdown/" target="_blank" rel="noopener">shortcodes</a>.</li>
+</ul>
+<p>Further event details, including <a href="https://docs.hugoblox.com/reference/markdown/" target="_blank" rel="noopener">page elements</a> such as image galleries, can be added to the body of this page.</p>
+
+    </div>
+
+    
+
+
+
+
+
+
+
+<div class="share-box">
+  <ul class="share">
+    
+      
+      
+      
+        
+      
+      
+      
+      
+      
+      
+      
+      <li>
+        <a href="https://twitter.com/intent/tweet?url=https%3A%2F%2Ffrankschae.github.io%2Ftalk%2Fexample-talk%2F&amp;text=Example&#43;Talk" target="_blank" rel="noopener" class="share-btn-twitter" aria-label="twitter">
+          <i class="fab fa-twitter"></i>
+        </a>
+      </li>
+    
+      
+      
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+      
+      
+      
+      
+      
+      
+      <li>
+        <a href="https://www.facebook.com/sharer.php?u=https%3A%2F%2Ffrankschae.github.io%2Ftalk%2Fexample-talk%2F&amp;t=Example&#43;Talk" target="_blank" rel="noopener" class="share-btn-facebook" aria-label="facebook">
+          <i class="fab fa-facebook"></i>
+        </a>
+      </li>
+    
+      
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+      
+      
+      
+        
+      
+      <li>
+        <a href="mailto:?subject=Example%20Talk&amp;body=https%3A%2F%2Ffrankschae.github.io%2Ftalk%2Fexample-talk%2F" target="_blank" rel="noopener" class="share-btn-email" aria-label="envelope">
+          <i class="fas fa-envelope"></i>
+        </a>
+      </li>
+    
+      
+      
+      
+        
+      
+      
+      
+      
+      
+      
+      
+      <li>
+        <a href="https://www.linkedin.com/shareArticle?url=https%3A%2F%2Ffrankschae.github.io%2Ftalk%2Fexample-talk%2F&amp;title=Example&#43;Talk" target="_blank" rel="noopener" class="share-btn-linkedin" aria-label="linkedin-in">
+          <i class="fab fa-linkedin-in"></i>
+        </a>
+      </li>
+    
+      
+      
+      
+        
+      
+      
+      
+      
+      
+      
+      
+      <li>
+        <a href="whatsapp://send?text=Example&#43;Talk%20https%3A%2F%2Ffrankschae.github.io%2Ftalk%2Fexample-talk%2F" target="_blank" rel="noopener" class="share-btn-whatsapp" aria-label="whatsapp">
+          <i class="fab fa-whatsapp"></i>
+        </a>
+      </li>
+    
+      
+      
+      
+        
+      
+      
+      
+      
+      
+      
+      
+      <li>
+        <a href="https://service.weibo.com/share/share.php?url=https%3A%2F%2Ffrankschae.github.io%2Ftalk%2Fexample-talk%2F&amp;title=Example&#43;Talk" target="_blank" rel="noopener" class="share-btn-weibo" aria-label="weibo">
+          <i class="fab fa-weibo"></i>
+        </a>
+      </li>
+    
+  </ul>
+</div>
+
+
+
+
+
+
+
+
+
+
+
+  
+  
+
+
+
+  
+  
+  
+    
+  
+  
+  
+  <div class="media author-card content-widget-hr">
+    
+      
+      <a href="https://frankschae.github.io/"><img class="avatar mr-3 avatar-circle" src="/authors/admin/avatar_hu5bdd272e648badb42a595cb37d3ba435_2457515_270x270_fill_lanczos_center_3.png" alt="Frank Schäfer"></a>
+    
+
+    <div class="media-body">
+      <h5 class="card-title"><a href="https://frankschae.github.io/">Frank Schäfer</a></h5>
+      <h6 class="card-subtitle">Postdoctoral researcher</h6>
+      <p class="card-text">My research interests include many-body physics, probabilistic machine learning, and differentiable programming.</p>
+      <ul class="network-icon" aria-hidden="true">
+  
+    
+    
+    
+      
+    
+    
+    
+    
+    
+      
+    
+    <li>
+      <a href="/#contact" >
+        <i class="fas fa-envelope"></i>
+      </a>
+    </li>
+  
+    
+    
+    
+      
+    
+    
+    
+    
+    
+      
+    
+    <li>
+      <a href="https://twitter.com/_Frank_Schaefer" target="_blank" rel="noopener">
+        <i class="fab fa-twitter"></i>
+      </a>
+    </li>
+  
+    
+    
+    
+    
+    
+    
+    
+      
+    
+    <li>
+      <a href="https://scholar.google.com/citations?user=goAokcEAAAAJ&amp;hl=de&amp;oi=sra#" target="_blank" rel="noopener">
+        <i class="ai ai-google-scholar"></i>
+      </a>
+    </li>
+  
+    
+    
+    
+      
+    
+    
+    
+    
+    
+      
+    
+    <li>
+      <a href="https://github.com/frankschae" target="_blank" rel="noopener">
+        <i class="fab fa-github"></i>
+      </a>
+    </li>
+  
+    
+    
+    
+    
+    
+    
+    
+      
+    
+    <li>
+      <a href="https://orcid.org/0000-0003-2684-4984" target="_blank" rel="noopener">
+        <i class="ai ai-orcid"></i>
+      </a>
+    </li>
+  
+</ul>
+
+    </div>
+  </div>
+
+
+
+
+
+
+
+
+
+
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+
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+
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+
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+  </div>
+</div>
+  </div>
+
+  <div class="page-footer">
+    
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+    <div class="container">
+      <footer class="site-footer">
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+  
+    
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+  
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+  
+  <p class="powered-by copyright-license-text">
+    © 2024 Frank Schäfer. This work is licensed under <a href="https://creativecommons.org/licenses/by-nc-nd/4.0" rel="noopener noreferrer" target="_blank">CC BY NC ND 4.0</a>
+  </p>
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+  <p class="powered-by">
+    
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+      
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+      
+      Published with <a href="https://hugoblox.com/?utm_campaign=poweredby" target="_blank" rel="noopener">Hugo Blox Builder</a> — the free, <a href="https://github.com/HugoBlox/hugo-blox-builder" target="_blank" rel="noopener">open source</a> website builder that empowers creators.
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+
+
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+  
+  <script id="search-hit-fuse-template" type="text/x-template">
+    <div class="search-hit" id="summary-{{key}}">
+      <div class="search-hit-content">
+        <div class="search-hit-name">
+          <a href="{{relpermalink}}">{{title}}</a>
+          <div class="article-metadata search-hit-type">{{type}}</div>
+          <p class="search-hit-description">{{snippet}}</p>
+        </div>
+      </div>
+    </div>
+  </script>
+  
+    <script src="https://cdn.jsdelivr.net/gh/krisk/Fuse@v3.2.1/dist/fuse.min.js" integrity="sha512-o38bmzBGX+hD3JHWUFCDA09btWaqrNmoJ3RXLlrysA7PP01Kgs4UlE4MhelE1v5dJR3+cxlR4qQlotsW7jKsnw==" crossorigin="anonymous"></script>
+    <script src="https://cdn.jsdelivr.net/gh/julmot/mark.js@8.11.1/dist/jquery.mark.min.js" integrity="sha512-mhbv5DqBMgrWL+32MmsDOt/OAvqr/cHimk6B8y/bx/xS88MVkYGPiVv2ixKVrkywF2qHplNRUvFsAHUdxZ3Krg==" crossorigin="anonymous"></script>
+  
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+<script id="page-data" type="application/json">{"use_headroom":true}</script>
+
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+  <script src="/js/wowchemy-headroom.db4755770454eb63685f8de785c0a172.js" type="module"></script>
+
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+  
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+<script src="/en/js/wowchemy.min.7f5ebaff62ae468cff8bb3dd1337bb9b.js"></script>
+
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+  <script src="/js/wowchemy-map.a26e9d2f7238ba5b868384f1c5bc6477.js" type="module"></script>
+
+
+
+
+  
+<div id="modal" class="modal fade" role="dialog">
+  <div class="modal-dialog">
+    <div class="modal-content">
+      <div class="modal-header">
+        <h5 class="modal-title">Cite</h5>
+        <button type="button" class="close" data-dismiss="modal" aria-label="Close">
+          <span aria-hidden="true">&times;</span>
+        </button>
+      </div>
+      <div class="modal-body">
+        
+        <pre><code></code></pre>
+      </div>
+      <div class="modal-footer">
+        <a class="btn btn-outline-primary my-1 js-copy-cite" href="#" target="_blank">
+          <i class="fas fa-copy"></i> Copy
+        </a>
+        <a class="btn btn-outline-primary my-1 js-download-cite" href="#" target="_blank">
+          <i class="fas fa-download"></i> Download
+        </a>
+        <div id="modal-error"></div>
+      </div>
+    </div>
+  </div>
+</div>
+
+
+  <script src="/js/wowchemy-publication.9c0e895144aef5a693008b5c5d450147.js" type="module"></script>
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+</body>
+</html>
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